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All questions of Divisibility and Remainders for GRE Exam
For all positive integers m and v, the expression m Θ v represents the remainder when m is divided by v. What is the value of ( (98 Θ 33) Θ 17 )− ( 98 Θ (33 Θ 17) ) ? - a)-10
- b)-2
- c)8
- d)13
Correct answer is option 'D'. Can you explain this answer?
For all positive integers m and v, the expression m Θ v represents the remainder when m is divided by v. What is the value of ( (98 Θ 33) Θ 17 )− ( 98 Θ (33 Θ 17) ) ?
a)
-10
b)
-2
c)
8
d)
13
|
Aditya Kumar answered |
First for ((98 Θ 33) Θ 17), determine 98 Θ 33, which equalsd to 32, since 32 is remainder when 98 divided with 33 ( 98 = 2(33) + 32 ).
Then determine 32 Θ 17 which equals to 15, since 15 is the remainder when 32 is divided by 17( 33= 1(17) +15)
Thus ( (98 Θ 33) Θ 17 ) = 15
Next, for (98 Θ (33 Θ 17)) , determine 33 Θ 17, which equalsd to 16, since16is remainder when33 divided with17 ( 33 = 1(17) + 16 ).
Then determine 98 Θ 16 which equals to 2, since 2 is the remainder when 98 is divided by 16 ( 98 = 6(16) + 2)
Thus (98 Θ (33 Θ 17)) = 2
Finally ((98 Θ 33) Θ 17)− (98 Θ (33 Θ 17)) = 15 - 2 = 13
What is the remainder when the positive integer x is divided by 8?1) When x is divided by 12, the remainder is 5. ?2) When x is divided by 18, the remainder is 11. ?- a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'D'. Can you explain this answer?
What is the remainder when the positive integer x is divided by 8?
1) When x is divided by 12, the remainder is 5. ?
2) When x is divided by 18, the remainder is 11. ?
a)
Exactly one of the statements can answer the question
b)
Both statements are required to answer the question
c)
Each statement can answer the question individually
d)
More information is required as the information provided is insufficient to answer the question
|
Sankar Desai answered |
To find the remainder when the positive integer x is divided by 8, we need to consider the given statements.
Statement 1: When x is divided by 12, the remainder is 5.
Statement 2: When x is divided by 18, the remainder is 11.
To determine the remainder when x is divided by 8, we need to find a common divisor of 8, 12, and 18. Let's analyze each statement separately:
Statement 1: When x is divided by 12, the remainder is 5.
If we consider the possible values of x, we can see that when x is 5, 17, 29, etc., the remainder when divided by 12 is 5. However, this statement does not provide any information about the remainder when x is divided by 8.
Statement 2: When x is divided by 18, the remainder is 11.
If we consider the possible values of x, we can see that when x is 11, 29, 47, etc., the remainder when divided by 18 is 11. Again, this statement does not provide any information about the remainder when x is divided by 8.
Combined Analysis:
Since neither statement provides any information about the remainder when x is divided by 8, we cannot determine the remainder based on the given information alone. Therefore, the correct answer is option D: More information is required as the information provided is insufficient to answer the question.
To find the remainder when x is divided by 8, we would need additional information or a direct statement about the remainder when x is divided by 8.
Statement 1: When x is divided by 12, the remainder is 5.
Statement 2: When x is divided by 18, the remainder is 11.
To determine the remainder when x is divided by 8, we need to find a common divisor of 8, 12, and 18. Let's analyze each statement separately:
Statement 1: When x is divided by 12, the remainder is 5.
If we consider the possible values of x, we can see that when x is 5, 17, 29, etc., the remainder when divided by 12 is 5. However, this statement does not provide any information about the remainder when x is divided by 8.
Statement 2: When x is divided by 18, the remainder is 11.
If we consider the possible values of x, we can see that when x is 11, 29, 47, etc., the remainder when divided by 18 is 11. Again, this statement does not provide any information about the remainder when x is divided by 8.
Combined Analysis:
Since neither statement provides any information about the remainder when x is divided by 8, we cannot determine the remainder based on the given information alone. Therefore, the correct answer is option D: More information is required as the information provided is insufficient to answer the question.
To find the remainder when x is divided by 8, we would need additional information or a direct statement about the remainder when x is divided by 8.
How many multiples of 7 are there between 14 and 140, inclusive?- a)15
- b)16
- c)17
- d)18
- e)19
Correct answer is option 'E'. Can you explain this answer?
How many multiples of 7 are there between 14 and 140, inclusive?
a)
15
b)
16
c)
17
d)
18
e)
19
|
Baishali Chavan answered |
To find the number of multiples of 7 between 14 and 140, we need to divide 140 by 7 and subtract the result of dividing 14 by 7, then add 1 to include the endpoint 140.
Steps:
1. Divide 140 by 7: 140/7 = 20
2. Divide 14 by 7: 14/7 = 2
3. Subtract 2 from 20: 20 - 2 = 18
4. Add 1: 18 + 1 = 19
Therefore, there are 19 multiples of 7 between 14 and 140, inclusive.
The correct answer is (e) 19.
Steps:
1. Divide 140 by 7: 140/7 = 20
2. Divide 14 by 7: 14/7 = 2
3. Subtract 2 from 20: 20 - 2 = 18
4. Add 1: 18 + 1 = 19
Therefore, there are 19 multiples of 7 between 14 and 140, inclusive.
The correct answer is (e) 19.
What is the remainder when the positive integer n is divided by the positive integer k, where k > 1?1) n = (k+1)32) k = 5 - a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'A'. Can you explain this answer?
What is the remainder when the positive integer n is divided by the positive integer k, where k > 1?
1) n = (k+1)3
2) k = 5
a)
Exactly one of the statements can answer the question
b)
Both statements are required to answer the question
c)
Each statement can answer the question individually
d)
More information is required as the information provided is insufficient to answer the question
|
Dipanjan Mehra answered |
It is not possible to determine the remainder when n is divided by k without knowing the values of n and k.
If n is an integer, when (2n + 2)2 is divided by 4 the remainder is- a)0
- b)1
- c)2
- d)3
- e)4
Correct answer is option 'A'. Can you explain this answer?
If n is an integer, when (2n + 2)2 is divided by 4 the remainder is
a)
0
b)
1
c)
2
d)
3
e)
4
|
Wizius Careers answered |
We first expand (2n + 2)2
(2n + 2)2 = 4n2 + 8n + 4
Factor 4 out.
= 4(n2 + 2n + 1)
(2n + 2)2 is divisible by 4 and the remainder is equal to 0. The Answer is A.
(2n + 2)2 = 4n2 + 8n + 4
Factor 4 out.
= 4(n2 + 2n + 1)
(2n + 2)2 is divisible by 4 and the remainder is equal to 0. The Answer is A.
If n is a positive integer, what is the remainder when (7(4n+3))(6n) is divided by 10? - a)2
- b)4
- c)6
- d)8
Correct answer is option 'D'. Can you explain this answer?
If n is a positive integer, what is the remainder when (7(4n+3))(6n) is divided by 10?
a)
2
b)
4
c)
6
d)
8
|
Dhruv Mehra answered |
Cyclicity of 7 is 4.
if n = 1, power of 7 will 7 and, the reminder will 3 if 7 is divided by 4
if n = 3, power of 7 will 15 and, the reminder will 3 if 15 is divided by 4
Last digit of 7^4 will 3, and last digit 6 power anything will 6
Now, 6*3 = 18/10 = 8 Reminder.
What is the remainder when 1044 × 1047 × 1050 × 1053 is divided by 33?- a)3
- b)27
- c)30
- d)21
- e)18
Correct answer is option 'C'. Can you explain this answer?
What is the remainder when 1044 × 1047 × 1050 × 1053 is divided by 33?
a)
3
b)
27
c)
30
d)
21
e)
18
|
Meera Rana answered |
You can solve this problem if you know this rule about remainders.
Let a number x divide the product of A and B.
The remainder will be the product of the remainders when x divides A and when x divides B.
Let a number x divide the product of A and B.
The remainder will be the product of the remainders when x divides A and when x divides B.
Using this rule,
The remainder when 33 divides 1044 is 21.
The remainder when 33 divides 1047 is 24.
The remainder when 33 divides 1050 is 27.
The remainder when 33 divides 1053 is 30.
The remainder when 33 divides 1044 is 21.
The remainder when 33 divides 1047 is 24.
The remainder when 33 divides 1050 is 27.
The remainder when 33 divides 1053 is 30.
∴ the remainder when 33 divides 1044 × 1047 × 1050 × 1053 is 21 × 24 × 27 × 30.
Note:
The remainder when a number is divided by a divisor 'd' will take values from 0 to (d - 1). It will not be equal to or more than 'd'
The remainder when a number is divided by a divisor 'd' will take values from 0 to (d - 1). It will not be equal to or more than 'd'
The value of 21 × 24 × 27 × 30 is more than 33.
When the value of the remainder is more than the divisor, the final remainder will be the remainder of dividing the product by the divisor.
i.e., the final remainder is the remainder when 33 divides 21 × 24 × 27 × 30.
When 33 divides 21 × 24 × 27 × 30, the remainder is 30.
When the value of the remainder is more than the divisor, the final remainder will be the remainder of dividing the product by the divisor.
i.e., the final remainder is the remainder when 33 divides 21 × 24 × 27 × 30.
When 33 divides 21 × 24 × 27 × 30, the remainder is 30.
When 20 is divided by the positive integer k, the remainder is k – 2. Which of the following is a possible value of k? - a)8
- b)9
- c)10
- d)11
Correct answer is option 'D'. Can you explain this answer?
When 20 is divided by the positive integer k, the remainder is k – 2. Which of the following is a possible value of k?
a)
8
b)
9
c)
10
d)
11
|
Kshitij Jain answered |
Try dividing 20 by all the options, get remainders or K-2 for all the options and find suitable value for k by subtracting 2 from the K or divisor every time.
When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y? - a)15
- b)20
- c)28
- d)35
Correct answer is option 'D'. Can you explain this answer?
When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?
a)
15
b)
20
c)
28
d)
35
|
|
Aditya Deshmukh answered |
When the positive integer x is divided by 5 and 7, the remainder is 3 and 4, respectively: x=5q+3x=5q+3 (x could be 3, 8, 13, 18, 23, ...) and x=7p+4x=7p+4 (x could be 4, 11, 18, 25, ...).
There is a way to derive general formula based on above two statements:
Divisor will be the least common multiple of above two divisors 5 and 7, hence 3535.
Remainder will be the first common integer in above two patterns, hence 1818 --> so, to satisfy both this conditions x must be of a type x=35m+18x=35m+18 (18, 53, 88, ...);
The same for y (as the same info is given about y): y=35n+18y=35n+18;
x−y=(35m+18)−(35n+18)=35(m−n)x−y=(35m+18)−(35n+18)=35(m−n) --> thus x-y must be a multiple of 35.
When m is divided by n, the remainder is 14. If m/n = 65.4 , what is the value of n? - a)14
- b)27
- c)35
- d)42
Correct answer is option 'C'. Can you explain this answer?
When m is divided by n, the remainder is 14. If m/n = 65.4 , what is the value of n?
a)
14
b)
27
c)
35
d)
42
|
|
Akash Khamkar answered |
(m-14)/n leaves remainder as zero.
Now m/n = 65.4 where 65 is whole number and 0.4 represents remainder wrt n.
i.e. 0.4 = 14/n
so n = 14/0.4 = 35
Now m/n = 65.4 where 65 is whole number and 0.4 represents remainder wrt n.
i.e. 0.4 = 14/n
so n = 14/0.4 = 35
If K is a positive integer less than 10 and N = 4,321 + K, what is the value of K?1) N is divisible by 3 ?2) N is divisible by 7 ?- a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'A'. Can you explain this answer?
If K is a positive integer less than 10 and N = 4,321 + K, what is the value of K?
1) N is divisible by 3 ?
2) N is divisible by 7 ?
a)
Exactly one of the statements can answer the question
b)
Both statements are required to answer the question
c)
Each statement can answer the question individually
d)
More information is required as the information provided is insufficient to answer the question
|
|
Sharmila Singh answered |
To find the value of K, we need to determine if N = 4,321 is divisible by K. Let's analyze each statement:
Statement 1: N is divisible by 3
- If N is divisible by 3, it means that the sum of its digits is divisible by 3.
- The sum of the digits of N = 4,321 is 4 + 3 + 2 + 1 = 10, which is not divisible by 3.
- Therefore, N is not divisible by 3, which means K cannot be 3.
- However, there are still other possible values of K that could satisfy the condition, as K could be any positive integer less than 10 excluding 3.
- This statement alone is not sufficient to determine the value of K.
Statement 2: N is divisible by 7
- If N is divisible by 7, it means that when N is divided by 7, the remainder is 0.
- N = 4,321 when divided by 7 gives a remainder of 4.
- Therefore, N is not divisible by 7, which means K cannot be 7.
- However, there are still other possible values of K that could satisfy the condition, as K could be any positive integer less than 10 excluding 7.
- This statement alone is not sufficient to determine the value of K.
Combining both statements:
- From statement 1, we know that K cannot be 3.
- From statement 2, we know that K cannot be 7.
- Therefore, the only possible value of K that satisfies both statements is K = 1.
- Both statements together are sufficient to determine the value of K.
Hence, the correct answer is option A: Exactly one of the statements can answer the question.
Statement 1: N is divisible by 3
- If N is divisible by 3, it means that the sum of its digits is divisible by 3.
- The sum of the digits of N = 4,321 is 4 + 3 + 2 + 1 = 10, which is not divisible by 3.
- Therefore, N is not divisible by 3, which means K cannot be 3.
- However, there are still other possible values of K that could satisfy the condition, as K could be any positive integer less than 10 excluding 3.
- This statement alone is not sufficient to determine the value of K.
Statement 2: N is divisible by 7
- If N is divisible by 7, it means that when N is divided by 7, the remainder is 0.
- N = 4,321 when divided by 7 gives a remainder of 4.
- Therefore, N is not divisible by 7, which means K cannot be 7.
- However, there are still other possible values of K that could satisfy the condition, as K could be any positive integer less than 10 excluding 7.
- This statement alone is not sufficient to determine the value of K.
Combining both statements:
- From statement 1, we know that K cannot be 3.
- From statement 2, we know that K cannot be 7.
- Therefore, the only possible value of K that satisfies both statements is K = 1.
- Both statements together are sufficient to determine the value of K.
Hence, the correct answer is option A: Exactly one of the statements can answer the question.
When positive integer x is divided by positive integer y, the remainder is 9. If x/y = 96.12, what is the value of y? - a)96
- b)75
- c)48
- d)25
Correct answer is option 'B'. Can you explain this answer?
When positive integer x is divided by positive integer y, the remainder is 9. If x/y = 96.12, what is the value of y?
a)
96
b)
75
c)
48
d)
25
|
|
Nandita Chauhan answered |
The remainder is 9 when x is divided by y, so x = yq + 9 for some positive integer q.
Dividing both side by y give x/y = q + 9/y
But x/y = 96.12 = 96 + 0.12
Equating the two expressions for x/y give q + 9/y = 96 + 0.12
Thus q = 96 and 9/y = 0.12
⇒ 9 = 0.12y
⇒ y = 9/0.12
⇒ y = 75
What is the number of even factors of 36000 which are divisible by 9 but not by 36?- a)20
- b)4
- c)10
- d)12
- e)16
Correct answer is option 'B'. Can you explain this answer?
What is the number of even factors of 36000 which are divisible by 9 but not by 36?
a)
20
b)
4
c)
10
d)
12
e)
16
|
|
Yash Rane answered |
36000=2^5 * 3^2 * 5^3
Since we are talking of even factors, there must be at least one 2 in the required factors.
Since the number is divisible by 9, we must have both the threes.
We cannot have more than 1 two as it will make the number divisible by 36.
So we have 1 way of choosing 2, 1 way of choosing 3, 4 ways of choosing 5.
Thus the required number of factors are
1*1*4 = 4
Since the number is divisible by 9, we must have both the threes.
We cannot have more than 1 two as it will make the number divisible by 36.
So we have 1 way of choosing 2, 1 way of choosing 3, 4 ways of choosing 5.
Thus the required number of factors are
1*1*4 = 4
Which of the following is a multiple of 6?
- a)414314
- b)512365
- c)618520
- d)726573
- e)878052
Correct answer is option 'E'. Can you explain this answer?
Which of the following is a multiple of 6?
a)
414314
b)
512365
c)
618520
d)
726573
e)
878052
|
Sameer Rane answered |
no must be divisible by 2 and 3
If a number N is divisible by both 2 and 8, then which of the following statements must be true?I. N is divisible by 4II. N is divisible by 6III. N is divisible by 16- a)I only
- b)II only
- c)III only
- d)I and II only
- e)I and III only
Correct answer is option 'A'. Can you explain this answer?
If a number N is divisible by both 2 and 8, then which of the following statements must be true?
I. N is divisible by 4
II. N is divisible by 6
III. N is divisible by 16
a)
I only
b)
II only
c)
III only
d)
I and II only
e)
I and III only
|
|
Stuti Dey answered |
Solution:
To solve this question, we need to use the concept of the LCM (Least Common Multiple) of the given numbers.
LCM of 2 and 8 is 8.
Therefore, any number that is divisible by both 2 and 8 must be a multiple of 8.
I. N is divisible by 4
If a number is divisible by 8, it is also divisible by 4. Therefore, statement I is true.
II. N is divisible by 6
If a number is divisible by 8, it is not necessarily divisible by 6. Therefore, statement II is false.
III. N is divisible by 16
If a number is divisible by 8, it is not necessarily divisible by 16. Therefore, statement III is false.
Therefore, the only statement that must be true is statement I, and the correct answer is option A.
To solve this question, we need to use the concept of the LCM (Least Common Multiple) of the given numbers.
LCM of 2 and 8 is 8.
Therefore, any number that is divisible by both 2 and 8 must be a multiple of 8.
I. N is divisible by 4
If a number is divisible by 8, it is also divisible by 4. Therefore, statement I is true.
II. N is divisible by 6
If a number is divisible by 8, it is not necessarily divisible by 6. Therefore, statement II is false.
III. N is divisible by 16
If a number is divisible by 8, it is not necessarily divisible by 16. Therefore, statement III is false.
Therefore, the only statement that must be true is statement I, and the correct answer is option A.
What is the remainder when the two-digit, positive integer x is divided by 3 ?1) The sum of the digits of x is 5.?2) The remainder when x is divided by 9 is 5. - a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'C'. Can you explain this answer?
What is the remainder when the two-digit, positive integer x is divided by 3 ?
1) The sum of the digits of x is 5.?
2) The remainder when x is divided by 9 is 5.
a)
Exactly one of the statements can answer the question
b)
Both statements are required to answer the question
c)
Each statement can answer the question individually
d)
More information is required as the information provided is insufficient to answer the question
|
|
Akshay Khanna answered |
To find the remainder when a two-digit positive integer x is divided by 3, we need to determine the value of x modulo 3.
Statement 1: The sum of the digits of x is 5.
This statement does not provide enough information to determine the remainder when x is divided by 3. For example, if x is 14, the remainder when divided by 3 is 2, but if x is 23, the remainder is 1. Therefore, statement 1 alone is insufficient.
Statement 2: The remainder when x is divided by 9 is 5.
This statement also does not provide enough information to determine the remainder when x is divided by 3. For example, if x is 23, the remainder when divided by 3 is 2, but if x is 32, the remainder is 1. Therefore, statement 2 alone is insufficient.
Combining both statements:
By considering both statements together, we can determine the value of x modulo 3. Since the remainder when x is divided by 9 is 5, we know that x is of the form 9k + 5, where k is an integer. Additionally, since the sum of the digits of x is 5, the possible values for x are 14, 23, 32, etc.
Analyzing the possible values of x modulo 3:
For x = 14, the remainder when divided by 3 is 2.
For x = 23, the remainder when divided by 3 is 2.
For x = 32, the remainder when divided by 3 is 2.
From the analysis above, we can see that regardless of the specific value of x, the remainder when x is divided by 3 is always 2. Therefore, by combining both statements, we can answer the question and statement 1 and 2 together are sufficient to answer the question.
Therefore, the correct answer is (C) Each statement can answer the question individually.
Statement 1: The sum of the digits of x is 5.
This statement does not provide enough information to determine the remainder when x is divided by 3. For example, if x is 14, the remainder when divided by 3 is 2, but if x is 23, the remainder is 1. Therefore, statement 1 alone is insufficient.
Statement 2: The remainder when x is divided by 9 is 5.
This statement also does not provide enough information to determine the remainder when x is divided by 3. For example, if x is 23, the remainder when divided by 3 is 2, but if x is 32, the remainder is 1. Therefore, statement 2 alone is insufficient.
Combining both statements:
By considering both statements together, we can determine the value of x modulo 3. Since the remainder when x is divided by 9 is 5, we know that x is of the form 9k + 5, where k is an integer. Additionally, since the sum of the digits of x is 5, the possible values for x are 14, 23, 32, etc.
Analyzing the possible values of x modulo 3:
For x = 14, the remainder when divided by 3 is 2.
For x = 23, the remainder when divided by 3 is 2.
For x = 32, the remainder when divided by 3 is 2.
From the analysis above, we can see that regardless of the specific value of x, the remainder when x is divided by 3 is always 2. Therefore, by combining both statements, we can answer the question and statement 1 and 2 together are sufficient to answer the question.
Therefore, the correct answer is (C) Each statement can answer the question individually.
How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3? - a)15
- b)16
- c)17
- d)18
Correct answer is option 'C'. Can you explain this answer?
How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3?
a)
15
b)
16
c)
17
d)
18
|
|
Advait Malik answered |
1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49
A total of 17 numbers are there which gives a remainder 1 when divided by 3
A total of 17 numbers are there which gives a remainder 1 when divided by 3
What is the remainder when the sum of the positive integers x and y is divided by 6?1) When x is divided by 6, the remainder is 3. ?2) When y is divided by 6, the remainder is 1. - a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'B'. Can you explain this answer?
What is the remainder when the sum of the positive integers x and y is divided by 6?
1) When x is divided by 6, the remainder is 3. ?
2) When y is divided by 6, the remainder is 1.
a)
Exactly one of the statements can answer the question
b)
Both statements are required to answer the question
c)
Each statement can answer the question individually
d)
More information is required as the information provided is insufficient to answer the question
|
Almir Mustafin answered |
(6n+3)/6+(6m+1)/6=n+1/2+m+1/6=n+m+2/3 of x+y.
What is the smallest integer that is multiple of 5, 7, and 20?- a)70
- b)35
- c)200
- d)280
- e)140
Correct answer is option 'E'. Can you explain this answer?
What is the smallest integer that is multiple of 5, 7, and 20?
a)
70
b)
35
c)
200
d)
280
e)
140
|
|
Hridoy Gupta answered |
Finding the smallest integer that is a multiple of 5, 7, and 20 requires finding their least common multiple (LCM).
METHOD 1: Prime Factorization
Step 1: Prime factorize each number
5 = 5
7 = 7
20 = 2^2 x 5
Step 2: Find the highest power of each prime factor
5: 5
7: 7
2: 2^2
Step 3: Multiply the prime factors raised to their highest power
5 x 7 x 2^2 = 140
Therefore, the smallest integer that is a multiple of 5, 7, and 20 is 140.
METHOD 2: Listing Multiples
Step 1: List the multiples of each number until you find a common multiple
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70
Multiples of 7: 7, 14, 21, 28, 35
Multiples of 20: 20, 40, 60, 80, 100, 120, 140
Step 2: Find the smallest common multiple
The smallest common multiple is 140.
Therefore, the smallest integer that is a multiple of 5, 7, and 20 is 140.
METHOD 1: Prime Factorization
Step 1: Prime factorize each number
5 = 5
7 = 7
20 = 2^2 x 5
Step 2: Find the highest power of each prime factor
5: 5
7: 7
2: 2^2
Step 3: Multiply the prime factors raised to their highest power
5 x 7 x 2^2 = 140
Therefore, the smallest integer that is a multiple of 5, 7, and 20 is 140.
METHOD 2: Listing Multiples
Step 1: List the multiples of each number until you find a common multiple
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70
Multiples of 7: 7, 14, 21, 28, 35
Multiples of 20: 20, 40, 60, 80, 100, 120, 140
Step 2: Find the smallest common multiple
The smallest common multiple is 140.
Therefore, the smallest integer that is a multiple of 5, 7, and 20 is 140.
If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?1) w + x + y + z = 132) N + 5 is divisible by 9- a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'C'. Can you explain this answer?
If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?
1) w + x + y + z = 13
2) N + 5 is divisible by 9
a)
Exactly one of the statements can answer the question
b)
Both statements are required to answer the question
c)
Each statement can answer the question individually
d)
More information is required as the information provided is insufficient to answer the question
|
|
Nayanika Bajaj answered |
GMAT Problem: If w, x, y, and z are the digits of the four-digit number N, a positive integer, what is the remainder when N is divided by 9?
Statement 1: w x y z = 13
If the sum of the digits of a number is divisible by 9, then the number itself is also divisible by 9. Here, the sum of the digits is 1+3+2=6, which is not divisible by 9. Therefore, this statement alone is not sufficient to answer the question.
Statement 2: N 5 is divisible by 9
If a number is divisible by 9, then the sum of its digits is also divisible by 9. Here, we know that N+5 is divisible by 9. Let's assume that N has digits a, b, c, and d. Then we have:
(a+b+c+d) + 5 = 9k, where k is some integer.
Simplifying, we get:
a+b+c+d = 9k-5
Since we want to find the remainder when N is divided by 9, we only need to know the value of (a+b+c+d) mod 9. From the equation above, we can see that (a+b+c+d) mod 9 is the same as (9k-5) mod 9. But 9k is always divisible by 9, so (9k-5) mod 9 is the same as (-5) mod 9, which is 4. Therefore, we know that (a+b+c+d) mod 9 is 4, and this statement alone is sufficient to answer the question.
Answer: Each statement can answer the question individually (Option C)
Statement 1: w x y z = 13
If the sum of the digits of a number is divisible by 9, then the number itself is also divisible by 9. Here, the sum of the digits is 1+3+2=6, which is not divisible by 9. Therefore, this statement alone is not sufficient to answer the question.
Statement 2: N 5 is divisible by 9
If a number is divisible by 9, then the sum of its digits is also divisible by 9. Here, we know that N+5 is divisible by 9. Let's assume that N has digits a, b, c, and d. Then we have:
(a+b+c+d) + 5 = 9k, where k is some integer.
Simplifying, we get:
a+b+c+d = 9k-5
Since we want to find the remainder when N is divided by 9, we only need to know the value of (a+b+c+d) mod 9. From the equation above, we can see that (a+b+c+d) mod 9 is the same as (9k-5) mod 9. But 9k is always divisible by 9, so (9k-5) mod 9 is the same as (-5) mod 9, which is 4. Therefore, we know that (a+b+c+d) mod 9 is 4, and this statement alone is sufficient to answer the question.
Answer: Each statement can answer the question individually (Option C)
For a nonnegative integer n, if the remainder is 1 when 2n is divided by 3, then which of the following must be true?I. n is greater than zero. ?
II. 3n = (-3)n ?
III. (√2)n is an integer. - a)I only
- b)II only
- c)II and III only
- d)I, II, and III
Correct answer is option 'C'. Can you explain this answer?
For a nonnegative integer n, if the remainder is 1 when 2n is divided by 3, then which of the following must be true?
I. n is greater than zero. ?
II. 3n = (-3)n ?
III. (√2)n is an integer.
II. 3n = (-3)n ?
III. (√2)n is an integer.
a)
I only
b)
II only
c)
II and III only
d)
I, II, and III
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EduRev GMAT answered |
If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r?1) 2 is not a factor of n. ?2) 3 is not a factor of n. - a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'B'. Can you explain this answer?
If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r?
1) 2 is not a factor of n. ?
2) 3 is not a factor of n.
a)
Exactly one of the statements can answer the question
b)
Both statements are required to answer the question
c)
Each statement can answer the question individually
d)
More information is required as the information provided is insufficient to answer the question
|
|
Advait Malik answered |
We can use the division algorithm to write:
n = dq + r
where d is a positive integer and 0 ≤ r < q.="" />
Rearranging this equation, we get:
r = n - dq
We know that n is a multiple of 13, so we can write:
n = 13k
where k is a positive integer.
Substituting this into the equation above, we get:
r = 13k - dq
Since r is less than q, we can write:
r = q - s
where s is a positive integer.
Substituting this into the equation above, we get:
q - s = 13k - dq
Simplifying, we get:
dq + s = 13k - q
Since d and q are positive integers, we know that dq + s is greater than q. Therefore, we can write:
dq + s = q + 13m
where m is a positive integer.
Substituting this into the equation above, we get:
q + 13m = 13k - q
Simplifying, we get:
2q + 13m = 13k
This shows that 2q is a multiple of 13. Since 13 is a prime number, this means that q must be a multiple of 13.
Therefore, the remainder r when n is divided by q must be less than q, and it must also be a multiple of 13. The only multiple of 13 that is less than q is 0. Therefore, the remainder r must be 0.
n = dq + r
where d is a positive integer and 0 ≤ r < q.="" />
Rearranging this equation, we get:
r = n - dq
We know that n is a multiple of 13, so we can write:
n = 13k
where k is a positive integer.
Substituting this into the equation above, we get:
r = 13k - dq
Since r is less than q, we can write:
r = q - s
where s is a positive integer.
Substituting this into the equation above, we get:
q - s = 13k - dq
Simplifying, we get:
dq + s = 13k - q
Since d and q are positive integers, we know that dq + s is greater than q. Therefore, we can write:
dq + s = q + 13m
where m is a positive integer.
Substituting this into the equation above, we get:
q + 13m = 13k - q
Simplifying, we get:
2q + 13m = 13k
This shows that 2q is a multiple of 13. Since 13 is a prime number, this means that q must be a multiple of 13.
Therefore, the remainder r when n is divided by q must be less than q, and it must also be a multiple of 13. The only multiple of 13 that is less than q is 0. Therefore, the remainder r must be 0.
When 15n, where n is a positive integer, is divided by 6, the remainder is x. What is the value of x?1) When n is divided by 2, the remainder is 0. ?2) When n is divided by 3, the remainder is 0. ?- a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'A'. Can you explain this answer?
When 15n, where n is a positive integer, is divided by 6, the remainder is x. What is the value of x?
1) When n is divided by 2, the remainder is 0. ?
2) When n is divided by 3, the remainder is 0. ?
a)
Exactly one of the statements can answer the question
b)
Both statements are required to answer the question
c)
Each statement can answer the question individually
d)
More information is required as the information provided is insufficient to answer the question
|
|
EduRev GMAT answered |
- From st1 we can know that n is a positive even number.
- ie2,4,6,8...
- So 15n/6 will always give the remainder 0. Sufficient
- From st2 n is a multiple of 3. For odd multiples of 3 the remainder of 15n/6 is 3. But for even multiple of 3 it’s 0. Two values. So insufficient
If an integer n is divisible by 3, 5, and 12, what is the next larger integer divisible by all these numbers?- a)n + 3
- b)n + 5
- c)n + 12
- d)n + 60
- e)n + 15
Correct answer is option 'D'. Can you explain this answer?
If an integer n is divisible by 3, 5, and 12, what is the next larger integer divisible by all these numbers?
a)
n + 3
b)
n + 5
c)
n + 12
d)
n + 60
e)
n + 15
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|
EduRev GMAT answered |
If n is divisible by 3, 5 and 12 it must a multiple of the lcm of 3, 5 and 12 which is 60.
n = 60 k
n + 60 is also divisible by 60 since
n + 60 = 60 k + 60 = 60(k + 1)
The answer is D
n = 60 k
n + 60 is also divisible by 60 since
n + 60 = 60 k + 60 = 60(k + 1)
The answer is D
If a positive integer n is divided by 5, the remainder is 3. Which of the numbers below yields a remainder of 0 when it is divided by 5? - a)n + 3
- b)n + 2
- c)n - 1
- d)n - 2
- e)n + 1
Correct answer is option 'B'. Can you explain this answer?
If a positive integer n is divided by 5, the remainder is 3. Which of the numbers below yields a remainder of 0 when it is divided by 5?
a)
n + 3
b)
n + 2
c)
n - 1
d)
n - 2
e)
n + 1
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|
Avantika Dey answered |
n divided by 5 yields a remainder equal to 3 is written as follows
n = 5 k + 3 , where k is an integer.
add 2 to both sides of the above equation to obtain
n + 2 = 5 k + 5 = 5(k + 1)
The above suggests that n + 2 divided by 5 yields a remainder equal to zero. The answer is B.
If an integer n is divisible by 3, 5 and 12, what is the next larger integer divisible by all these numbers? - a)n + 3
- b)n + 5
- c)n + 12
- d)n + 60
- e)n + 15
Correct answer is option 'D'. Can you explain this answer?
If an integer n is divisible by 3, 5 and 12, what is the next larger integer divisible by all these numbers?
a)
n + 3
b)
n + 5
c)
n + 12
d)
n + 60
e)
n + 15
|
|
Amar Chakraborty answered |
If n is divisible by 3, 5 and 12 it must a multiple of the lcm of 3, 5 and 12 which is 60.
n = 60 k
n + 60 is also divisible by 60 since
n + 60 = 60 k + 60 = 60(k + 1)
The answer is D.
n = 60 k
n + 60 is also divisible by 60 since
n + 60 = 60 k + 60 = 60(k + 1)
The answer is D.
The remainder when the positive integer m is divided by n is r. What is the remainder when 2m is divided by 2n? - a)r
- b)2r
- c)2n
- d)m - nr
Correct answer is option 'B'. Can you explain this answer?
The remainder when the positive integer m is divided by n is r. What is the remainder when 2m is divided by 2n?
a)
r
b)
2r
c)
2n
d)
m - nr
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|
Sinjini Mukherjee answered |
Plug in some numebrs and check - m = 5 n = 3 r = 2 What is the remainder when 2m is divided by 2n ? 2m = 10 2n = 6 So, Remainder when 2m is divided by 2n is = 4 2r = 4 Thus, answer must be 2r
What is the remainder when the positive integer n is divided by 3?1) The remainder when n is divided by 2 is 1.?2) The remainder when n + 1 is divided by 3 is 2.- a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'A'. Can you explain this answer?
What is the remainder when the positive integer n is divided by 3?
1) The remainder when n is divided by 2 is 1.?
2) The remainder when n + 1 is divided by 3 is 2.
a)
Exactly one of the statements can answer the question
b)
Both statements are required to answer the question
c)
Each statement can answer the question individually
d)
More information is required as the information provided is insufficient to answer the question
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Pallabi Deshpande answered |
(1) The remainder when n is divided by 2 is 1.
n is odd, some odd nos are evenly divisble by 3, some are not INSUFF
(2) The remainder when n + 1 is divided by 3 is 2.
n+1 = 3a +2
n =3a +1
so remainder is 1
SUFF
If t is a positive integer, can t2 + 1 be evenly divided by 10?(1) 916 × t leaves a remainder of 1 when divided by 2(2) 916 × t leaves a remainder of 2 when divided by 5- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- c)BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- d)EACH statement ALONE is sufficient.
- e)Statements (1) and (2) TOGETHER are NOT sufficient.
Correct answer is option 'C'. Can you explain this answer?
If t is a positive integer, can t2 + 1 be evenly divided by 10?
(1) 916 × t leaves a remainder of 1 when divided by 2
(2) 916 × t leaves a remainder of 2 when divided by 5
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
c)
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
d)
EACH statement ALONE is sufficient.
e)
Statements (1) and (2) TOGETHER are NOT sufficient.
|
|
Manasa Menon answered |
Steps 1 & 2: Understand Question and Draw Inferences
t2 + 1 can be evenly divided by 10 if t2 + 1 = 10 *k, where k is a positive integer
- We need to check if the last digit of t2 + 1 is 0
Step 3: Analyze Statement 1
916 × t leaves a remainder of 1 when divided by 2
- t is odd
- last digit of t can be 1, 3, 5, 7 or 9
Not Sufficient.
Step 4: Analyze Statement 2
916 × t leaves a remainder of 2 when divided by 5
- last digit of 916 × t can be 2 or 7
- last digit of t can be 2 or 7
Not Sufficient
Step 5: Analyze Both Statements Together (if needed)
Inference from statement 1: last digit of t can be 1, 3, 5, 7 or 9
Inference from statement 2: last digit of t can be 2 or 7
Inference from statement 1 and statement 2: last digit of t can be 7
--> Last digit of t2 + 1 = Last digit of (9 + 1) = 0
Hence, t2 + 1 can be evenly divided by 10
Statement 1 and Statement 2 together are sufficient to answer the question.
Answer: Option (C)
X = a.bcIf a, b and c denote the units, tenths and hundredths digits in the decimal representation of X above, what is the value of the product abc? (1) 100X divided by 50 leaves a remainder 24 (2) 1000X divided by 8 leaves a remainder 0- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'E'. Can you explain this answer?
X = a.bc
If a, b and c denote the units, tenths and hundredths digits in the decimal representation of X above, what is the value of the product abc?
(1) 100X divided by 50 leaves a remainder 24
(2) 1000X divided by 8 leaves a remainder 0
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
|
|
Prisha Mukherjee answered |
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
- 100X divided by 50 leaves a remainder 24
- So, we can write: 100X=50k+24
- , where k is an integer . . . (I)
- From the given expression of X, we can write:
- 100X=100a+10b+c . . . (II)
- The units digit on left hand side of the equation = c
- The units digit on right hand side of the equation = 4
- This means, c = 4
- As a multiple of 5 always has its units digit equal to either 0 or 5, 5k + 2 will always have its units digit equal to 2 or 7. So, b = {2, 7}.
- However, a can have many more values.
- Thus, we only know the definite value of c
- But we do not know definite values of a and b
- Thus, we know a definite value of c
- But we do not know definite values of a and b
- So, we will not be able to find a definite value of the product a×b×c
Statement 1 is not sufficient to answer the question.
Step 4: Analyze Statement 2 independently
-
- 1000a is always divisible by 8 as 1000 is divisible by 8
- 100b will be divisible by 8, only if b is even
- b= {0, 2, 4, 6, 8}
- 10c will be divisible by 8 only if c = {4, 8}
- As we do not have unique values of a, b and c we cannot calculate the unique value of product of a, b and c
Therefore, Statement 2 is clearly not sufficient
Step 5: Analyze Both Statements Together (if needed)
- From Statement 1: c = 4 and 10a+b=5k+2
- , where k is an integer
- From Statement 2: b = {0, 2, 4, 6, 8} and c = {4, 8}
So even after combining both statements, we still don’t know the definite values of a and b, and therefore cannot find a unique value of the product a×b×c
The two statements together are not sufficient to answer the question.
Answer: Option E
If a positive integer n is divided by 5, the remainder is 3. Which of the numbers below yields a remainder of 0 when it is divided by 5?- a) n + 3
- b) n + 2
- c)n - 1
- d) n - 2
- e)n + 1
Correct answer is option 'B'. Can you explain this answer?
If a positive integer n is divided by 5, the remainder is 3. Which of the numbers below yields a remainder of 0 when it is divided by 5?
a)
n + 3
b)
n + 2
c)
n - 1
d)
n - 2
e)
n + 1
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|
Gitanjali Kumar answered |
To find the number that yields a remainder of 0 when divided by 5, we need to understand the concept of remainders and how they are affected by addition and subtraction.
Let's assume the positive integer n is divided by 5, and the remainder is 3. This can be expressed as:
n = 5a + 3
where a is a positive integer representing the quotient.
To find a number that yields a remainder of 0 when divided by 5, we need to find a value of n that satisfies the equation:
n = 5b + 0
where b is a positive integer representing the quotient.
Let's analyze the answer options:
a) n + 3: Adding 3 to the number n will not change the remainder when divided by 5. It will still be 3, so this option does not yield a remainder of 0.
b) n + 2: Adding 2 to the number n will increase the remainder by 2 when divided by 5. Since the remainder was originally 3, adding 2 will result in a remainder of 0. This option satisfies the condition and yields a remainder of 0.
c) n - 1: Subtracting 1 from the number n will decrease the remainder by 1 when divided by 5. Since the remainder was originally 3, subtracting 1 will result in a remainder of 2. This option does not yield a remainder of 0.
d) n - 2: Subtracting 2 from the number n will decrease the remainder by 2 when divided by 5. Since the remainder was originally 3, subtracting 2 will result in a remainder of 1. This option does not yield a remainder of 0.
e) n + 1: Adding 1 to the number n will increase the remainder by 1 when divided by 5. Since the remainder was originally 3, adding 1 will result in a remainder of 4. This option does not yield a remainder of 0.
Therefore, the only option that yields a remainder of 0 when divided by 5 is option b) n + 2.
Let's assume the positive integer n is divided by 5, and the remainder is 3. This can be expressed as:
n = 5a + 3
where a is a positive integer representing the quotient.
To find a number that yields a remainder of 0 when divided by 5, we need to find a value of n that satisfies the equation:
n = 5b + 0
where b is a positive integer representing the quotient.
Let's analyze the answer options:
a) n + 3: Adding 3 to the number n will not change the remainder when divided by 5. It will still be 3, so this option does not yield a remainder of 0.
b) n + 2: Adding 2 to the number n will increase the remainder by 2 when divided by 5. Since the remainder was originally 3, adding 2 will result in a remainder of 0. This option satisfies the condition and yields a remainder of 0.
c) n - 1: Subtracting 1 from the number n will decrease the remainder by 1 when divided by 5. Since the remainder was originally 3, subtracting 1 will result in a remainder of 2. This option does not yield a remainder of 0.
d) n - 2: Subtracting 2 from the number n will decrease the remainder by 2 when divided by 5. Since the remainder was originally 3, subtracting 2 will result in a remainder of 1. This option does not yield a remainder of 0.
e) n + 1: Adding 1 to the number n will increase the remainder by 1 when divided by 5. Since the remainder was originally 3, adding 1 will result in a remainder of 4. This option does not yield a remainder of 0.
Therefore, the only option that yields a remainder of 0 when divided by 5 is option b) n + 2.
A function D(a, 10b + c) is defined as the remainder when the sum a0 + a1….a10b + c is divided by c, where a, b and c are single-digit positive integers.What is the value of D(y, 10x + z) where x, y and z are single-digit positive integers such that x < y < z , x and z are perfect squares and the difference between the sum and the product of the prime factors of y is 1?- a)0
- b)3
- c)6
- d)7
- e)Cannot be determined
Correct answer is option 'D'. Can you explain this answer?
A function D(a, 10b + c) is defined as the remainder when the sum a0 + a1….a10b + c is divided by c, where a, b and c are single-digit positive integers.
What is the value of D(y, 10x + z) where x, y and z are single-digit positive integers such that x < y < z , x and z are perfect squares and the difference between the sum and the product of the prime factors of y is 1?
a)
0
b)
3
c)
6
d)
7
e)
Cannot be determined
|
|
Sankar Desai answered |
Given
- D(a, 10b + c) =remainder when (a0 + a1….a10b + c ) is divided by c
- a, b and c are single-digit integers > 0
- x, y , z are single digit integers > 0
- x < y < z
- x = p2 and z = q2, where p, q are integers
- If y has only one prime factor then, then the difference between the product and the sum of prime factors would be 0
- So, we can say that y has more than 1 prime factor
To Find: Value of D(y, 10x + z)
- That is, the value of the remainder when y0+y1…..y10x+z is divided by z
- Approach
- For finding the remainder, we need to find the value of x, y and z
- Finding value of y
- We have inferred that y has more than one 1 prime factor.
- Also, we will consider the possible combinations of 2 or more prime factors keeping in the mind the constraint that y lies between 2 distinct single digit perfect squares
- There are 3 single digit perfect squares = {1, 4, 9}
- Finding values of x and z
- We are given that x =p2 and z =q2. Also, as x and z are integers < 10, the perfect squares less than 10 can be 1, 4 or 9.
- We will apply the constraints given on x and z to determine their values.
- Working Out
- Finding value of y
- We have inferred that y has more than one prime factor.
- Now, the smallest prime factors are {2, 3, 5, 7…}
- We will see which combinations of the above prime factors results in a value of y < 10
- From the above set, we can see that there is only one possible combination for which the value of y < 10, i.e. y = 2* 3 = 6
- So, the product of prime factors of y = 6 and the sum of prime factors of y = 5.
- Therefore the difference between the product and the sum of prime factors of y = 1
- Hence y = 6
- Finding values of x and z
- Perfect squares less than 10 = {1, 4, 9}. We are not assuming 0 here, as we are given that x, y and z are positive.
- As x < z, the possible values of (x, z) = (1, 4) , (1, 9) or (4, 9)
- Since y = 6 and x < y < z, (x, z) ≠ (1, 4)
- So, (x, z) = (1, 9) or (4, 9)
3. Let’s find the value of D(y, 10x + z) for both the cases of values of (x, z)
a. Case-I: y = 6 and (x, z) = (1, 9)
- 60+61……619 is to be divided by 9 = 32
- Since 6=2∗3,62=22∗32
- So, all the powers of 6 greater than 1, will be divisible by 9
- Therefore D(6, 10 +9) = Remainder when 60+61(=1+6=7)
- is divided by 9 = 7
- So, D(6,10 + 9) =
b. Case-II: y = 6 and (x, z) = (4, 9)
- 60+61……649 is to be divided by 9 = 32
- Since 6=2∗3,62=22∗32
- So, all the powers of 6 greater than 1, will be divisible by 9
- Therefore D(6, 40 +9) = Remainder when 60+61
- (= 1 + 6 + 7) is divided by 9 = 7
- So, D(6, 10 + 9) = 7
- 4. We see that in both the cases, the value of D(x, 10y+z) = 7.Answer: D
How many three digit numbers are divisible by 5 or 9?- a)260
- b)280
- c)200
- d)180...
- e)220
Correct answer is option 'A'. Can you explain this answer?
How many three digit numbers are divisible by 5 or 9?
a)
260
b)
280
c)
200
d)
180...
e)
220
|
|
Kiran Chauhan answered |
Three digit numbers divisible by 5 or 9 = three digit numbers divisible by 5 + three digit numbers divisible by 9 – three digit numbers divisible by 5 and 9.
The three digit numbers divisible by 5 = 100, 105, 110….995
The sequence given is in A.P with common difference 5. Let 995 be the nth term of the A.P, then
995 = 100 + (n – 1)5 = 100 + 5n – 5
Thus, n = 180 – (1)
The three digit numbers divisible by 9 = 108, 118, … 999
The sequence given is in A.P with common difference 9. Let 999 be the pth term of the A.P, then
999 = 108 + (p – 1)9 = 108 + 9p – 9
Thus, p = 100 – (2)
The three digit numbers divisible by 45 = 135, 180, …990
The sequence given is in A.P with common difference 45. Let 990 be the qth term of the A.P, then
990 = 135 + (q – 1)45 = 135 + 45q – 45
Thus, q = 20 – (3)
Thus, from (1), (2) and (3) the three digit numbers divisible by 5 or 9 = 180 + 100 – 20 = 260
The three digit numbers divisible by 5 = 100, 105, 110….995
The sequence given is in A.P with common difference 5. Let 995 be the nth term of the A.P, then
995 = 100 + (n – 1)5 = 100 + 5n – 5
Thus, n = 180 – (1)
The three digit numbers divisible by 9 = 108, 118, … 999
The sequence given is in A.P with common difference 9. Let 999 be the pth term of the A.P, then
999 = 108 + (p – 1)9 = 108 + 9p – 9
Thus, p = 100 – (2)
The three digit numbers divisible by 45 = 135, 180, …990
The sequence given is in A.P with common difference 45. Let 990 be the qth term of the A.P, then
990 = 135 + (q – 1)45 = 135 + 45q – 45
Thus, q = 20 – (3)
Thus, from (1), (2) and (3) the three digit numbers divisible by 5 or 9 = 180 + 100 – 20 = 260
A positive number is divided by 100 to get a remainder thrice as the quotient. If the number is divisible by 11, then how many such numbers are possible that are less than 100000?
Correct answer is '3'. Can you explain this answer?
A positive number is divided by 100 to get a remainder thrice as the quotient. If the number is divisible by 11, then how many such numbers are possible that are less than 100000?
|
|
Kiran Chauhan answered |
Let the number be N and the quotient when divided by 100 be k. Then remainder is 3k. 3k < 100
N = 100k + 3k = 103k,
Also, N is divisible by 11.
Also, N is divisible by 11.
=> k = 11p, where p is an integer.
=> N = 103*11p = 1133p...
As N < 100000, it implies that p can range from 1 to [100000/1133] i.e between 1 and 88
=> So, p can can range from 1 to 88
=> So, p can can range from 1 to 88
Also 3k < 100 => 3 * 11p < 100 => p < 4 Hence, p can take values 1,2,3
When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?- a)2
- b)4
- c)8
- d)10
- e)20
Correct answer is option 'B'. Can you explain this answer?
When a class of n students is divided into groups of 6 students each, 2 students are left without a group. When the class is divided into groups of 8 students each, 4 students are left without a group. What is the smallest number of students that can be added to or removed from the class so that the resulting number of students can be equally divided into groups of 12 students each?
a)
2
b)
4
c)
8
d)
10
e)
20
|
|
Krithika Datta answered |
Given:
- n = 6k + 2 = 8p + 4, where k and p are non-negative integers (if k or p were negative, then n would be negative too and that’s not possible since the number of students cannot be negative)
To find: The smallest number that can be added to or subtracted from n to make the resulting number divisible by 12
Approach:
1. Let the number to be added or subtracted from n be x. This means,
- Either (students are added) n + x = 12j, where j is a positive integer
- This means, n = 12j - x
- Or (students are removed) n – x = 12m, where m is a positive integer
- This means, n = 12m + x
2. The expressions n = 12j – x and n = 12m + x make us realize that in order to answer this question, we need to know the remainder when n is divided by 12
- Say the remainder is 3 (That is, n = 12m + 3). In this case, the smallest change to make the number of students divisible by 12 is to remove 3 students from the class.
- Same will be the case for any value of remainder less than or equal to 6
- Say the remainder is 10 (That is, n = 12m + 10). In this case, the smallest change to make the number of students divisible by 12 is to add 2 students to the class
- Same will be the case for any value of remainder greater than 6
3. Let the remainder that n leaves when divided by 12 be r. So, our Goal expression is: n = 12q + r, where quotient q is an integer and 0 ≤ r < 12
4. We’ll use the given information about n to drive towards our Goal expression
Working Out:
- n = 6k + 2 = 8p + 4
- LCM (6, 8) = 24
- This means, n is a number of the form 24z + r’, where 0 ≤ r’ < 24
- When 24z + r’ is divided by 6, the remainder is 2
- In this expression, the term 24z is divisible by 6
- So, possible values of r’ that can lead to remainder 2 = {2, 8, 14, 20}
- When 24z + r’ is divided by 6, the remainder is 2
- This means, n is a number of the form 24z + r’, where 0 ≤ r’ < 24
- LCM (6, 8) = 24
- When 24z + r’ is divided by 8, the remainder is 4
- In this expression, the term 24z is divisible by 8
- So, possible values of r’ that can lead to remainder 4 = {4, 12, 20}
- The only value of r’ that satisfies both these conditions = {20}
- So, r’ = 20
- Therefore, n = 24z +20
- Let’s now rearrange the above equation a little bit to make it comparable to our GOAL expression: n = {24z + 12} + 8
- Now this form is exactly comparable to our Goal Expression
- By comparison, we see that Remainder = 8
- Since remainder is greater than 6, the smallest change to make the number of students divisible by 12 is to add 4 students.
Looking at the answer choices, we see that the correct answer is Option B
If n is an integer, when (2n + 2)2 is divided by 4 the remainder is - a)0
- b)1
- c)2
- d)3
- e)4
Correct answer is option 'A'. Can you explain this answer?
If n is an integer, when (2n + 2)2 is divided by 4 the remainder is
a)
0
b)
1
c)
2
d)
3
e)
4
|
|
Nandita Yadav answered |
We first expand (2n + 2)2
(2n + 2)2 = 4n2 + 8n + 4
Factor 4 out.
= 4(n2 + 2n + 1)
(2n + 2)2 is divisible by 4 and the remainder is equal to 0. The answer is A.
(2n + 2)2 = 4n2 + 8n + 4
Factor 4 out.
= 4(n2 + 2n + 1)
(2n + 2)2 is divisible by 4 and the remainder is equal to 0. The answer is A.
n = 234yznis a positive integer whose tens and units digits are y and z respectively. It is given that n is divisible by 4, 5 and 9. Find n.- a)23400
- b)23401
- c)23410
- d)23411
- e)23412
Correct answer is option 'A'. Can you explain this answer?
n = 234yzn
is a positive integer whose tens and units digits are y and z respectively. It is given that n is divisible by 4, 5 and 9. Find n.
a)
23400
b)
23401
c)
23410
d)
23411
e)
23412
|
|
Mayank Joshi answered |
Given, n = 234yzn
- The number is divisible by 4 if the last two digits of n are divisible by 4.
- The number is divisible by 5 if the last digit of n is either 0 or 5.
- The number is divisible by 9 if the sum of the digits of n is divisible by 9.
Using the above information, we can deduce the following:
- Since n is divisible by 5, the last digit of n must be 0 or 5.
- Since n is divisible by 9, the sum of its digits is divisible by 9.
- Since y and z are the tens and units digits of n respectively, we can write n as:
n = 23400 + 10y + z
Now, we can use the divisibility rule for 9 to get:
2 + 3 + 4 + 0 + 0 + y + z = 9k
9 + y + z = 9k
y + z = 9(k-1)
Since y and z are digits, their sum can be at most 18. Therefore, k must be 2, which gives:
y + z = 9
We also know that n is divisible by 4, so the last two digits of n must be divisible by 4. This means that the number formed by yz must be divisible by 4. The possible pairs of digits that satisfy this condition and y + z = 9 are:
45, 81
Out of these, only 45 satisfies n = 23400 + 10y + z. Therefore, we have:
n = 23400 + 45
n = 23445
Hence, the correct answer is option A (23400).
- The number is divisible by 4 if the last two digits of n are divisible by 4.
- The number is divisible by 5 if the last digit of n is either 0 or 5.
- The number is divisible by 9 if the sum of the digits of n is divisible by 9.
Using the above information, we can deduce the following:
- Since n is divisible by 5, the last digit of n must be 0 or 5.
- Since n is divisible by 9, the sum of its digits is divisible by 9.
- Since y and z are the tens and units digits of n respectively, we can write n as:
n = 23400 + 10y + z
Now, we can use the divisibility rule for 9 to get:
2 + 3 + 4 + 0 + 0 + y + z = 9k
9 + y + z = 9k
y + z = 9(k-1)
Since y and z are digits, their sum can be at most 18. Therefore, k must be 2, which gives:
y + z = 9
We also know that n is divisible by 4, so the last two digits of n must be divisible by 4. This means that the number formed by yz must be divisible by 4. The possible pairs of digits that satisfy this condition and y + z = 9 are:
45, 81
Out of these, only 45 satisfies n = 23400 + 10y + z. Therefore, we have:
n = 23400 + 45
n = 23445
Hence, the correct answer is option A (23400).
If t is a positive integer and 8t is divisible by 96, what will be the remainder when t3 is divided by 108?- a)0
- b)5
- c)10
- d)15
- e)Cannot be determined
Correct answer is option 'A'. Can you explain this answer?
If t is a positive integer and 8t is divisible by 96, what will be the remainder when t3 is divided by 108?
a)
0
b)
5
c)
10
d)
15
e)
Cannot be determined
|
|
Manasa Menon answered |
Given: 108 = 22 × 33
Since 8t is divisible by 96, we may write
8t = 96k, where k is a positive integer
- 23t = 25 × 3× k
- t = 22 × 3× k
- t3 = 26 × 33 × k3
- t3 = ( 22 × 33 )( 24 × k3)
- t3 = 108( 24 × k3)
This shows that t3 is completely divisible by 108, implying that the remainder is 0.
Answer: Option (A)
How many positive integer values of N are possible if 21 is divisible by N?- a)3
- b)4
- c)5
- d)6
- e)7
Correct answer is option 'B'. Can you explain this answer?
How many positive integer values of N are possible if 21 is divisible by N?
a)
3
b)
4
c)
5
d)
6
e)
7
|
|
Kavya Nair answered |
Solution:
To find out the number of positive integer values of N, we need to find out all the factors of 21.
Factors of 21 are: 1, 3, 7, 21
Therefore, there are 4 positive integer values of N for which 21 is divisible. Hence, the correct option is (B).
To find out the number of positive integer values of N, we need to find out all the factors of 21.
Factors of 21 are: 1, 3, 7, 21
Therefore, there are 4 positive integer values of N for which 21 is divisible. Hence, the correct option is (B).
For positive integers a and b, a/b = 0.6 . Which of the following CANNOT be the value of a? - a)42
- b)105
- c)137
- d)174
Correct answer is option 'C'. Can you explain this answer?
For positive integers a and b, a/b = 0.6 . Which of the following CANNOT be the value of a?
a)
42
b)
105
c)
137
d)
174
|
|
Aditya Sharma answered |
Given:
- a and b are positive integers
- a/b = 0.6
To find:
Which of the following CANNOT be the value of a?
Solution:
To solve this problem, we need to understand the concept of rational numbers and their decimal representations. A rational number is any number that can be expressed as a fraction, where the numerator and denominator are both integers.
The decimal representation of a rational number can be terminating (the decimal representation ends) or non-terminating and repeating (the decimal representation has a repeating pattern). For example, 0.6 is a terminating decimal because it ends after one decimal place.
To express a fraction as a decimal, we divide the numerator by the denominator. In this case, a/b = 0.6.
Step 1: Express 0.6 as a fraction
To express 0.6 as a fraction, we need to determine the denominator that will result in a terminating decimal. Since 0.6 has one decimal place, the denominator should be a power of 10. Therefore, we can rewrite 0.6 as 6/10 or simplify it to 3/5.
Step 2: Analyze the answer choices
Now, let's analyze each answer choice to determine if it can be the value of a.
a) 42
To check if 42 can be the value of a, we divide 42 by 5. The result is 8.4, which is not equal to 0.6. Therefore, a = 42 can be a valid value.
b) 105
To check if 105 can be the value of a, we divide 105 by 5. The result is 21, which is not equal to 0.6. Therefore, a = 105 can be a valid value.
c) 137
To check if 137 can be the value of a, we divide 137 by 5. The result is 27.4, which is not equal to 0.6. Therefore, a = 137 can be a valid value.
d) 174
To check if 174 can be the value of a, we divide 174 by 5. The result is 34.8, which is not equal to 0.6. Therefore, a = 174 can be a valid value.
Conclusion:
Out of the given answer choices, the only value that cannot be the value of a is 137. Therefore, the correct answer is option C.
- a and b are positive integers
- a/b = 0.6
To find:
Which of the following CANNOT be the value of a?
Solution:
To solve this problem, we need to understand the concept of rational numbers and their decimal representations. A rational number is any number that can be expressed as a fraction, where the numerator and denominator are both integers.
The decimal representation of a rational number can be terminating (the decimal representation ends) or non-terminating and repeating (the decimal representation has a repeating pattern). For example, 0.6 is a terminating decimal because it ends after one decimal place.
To express a fraction as a decimal, we divide the numerator by the denominator. In this case, a/b = 0.6.
Step 1: Express 0.6 as a fraction
To express 0.6 as a fraction, we need to determine the denominator that will result in a terminating decimal. Since 0.6 has one decimal place, the denominator should be a power of 10. Therefore, we can rewrite 0.6 as 6/10 or simplify it to 3/5.
Step 2: Analyze the answer choices
Now, let's analyze each answer choice to determine if it can be the value of a.
a) 42
To check if 42 can be the value of a, we divide 42 by 5. The result is 8.4, which is not equal to 0.6. Therefore, a = 42 can be a valid value.
b) 105
To check if 105 can be the value of a, we divide 105 by 5. The result is 21, which is not equal to 0.6. Therefore, a = 105 can be a valid value.
c) 137
To check if 137 can be the value of a, we divide 137 by 5. The result is 27.4, which is not equal to 0.6. Therefore, a = 137 can be a valid value.
d) 174
To check if 174 can be the value of a, we divide 174 by 5. The result is 34.8, which is not equal to 0.6. Therefore, a = 174 can be a valid value.
Conclusion:
Out of the given answer choices, the only value that cannot be the value of a is 137. Therefore, the correct answer is option C.
What is the smallest positive 2-digit whole number divisible by 3 and such that the sum of its digits is 9? - a)27
- b)33
- c)72
- d)18
- e)90
Correct answer is option 'D'. Can you explain this answer?
What is the smallest positive 2-digit whole number divisible by 3 and such that the sum of its digits is 9?
a)
27
b)
33
c)
72
d)
18
e)
90
|
|
Gauri Iyer answered |
Let xy be the whole number with x and y the two digits that make up the number. The number is divisible by 3 may be written as follows
10 x + y = 3 k
The sum of x and y is equal to 9.
x + y = 9
Solve the above equation for y
y = 9 - x Substitute y = 9 - x in the equation 10 x + y = 3 k to obtain.
10 x + 9 - x = 3 k
Solve for x
x = (k - 3) / 3
x is a positive integer smaller than 10
Let k = 1, 2, 3, ... and select the first value that gives x as an integer. k = 6 gives x = 1
Find y using the equation y = 9 - x = 8
The number we are looking for is 18 and the answer is D. It is divisible by 3 and the sum of its digits is equal to 9 and it is the smallest and positive whole number with such properties.
10 x + y = 3 k
The sum of x and y is equal to 9.
x + y = 9
Solve the above equation for y
y = 9 - x Substitute y = 9 - x in the equation 10 x + y = 3 k to obtain.
10 x + 9 - x = 3 k
Solve for x
x = (k - 3) / 3
x is a positive integer smaller than 10
Let k = 1, 2, 3, ... and select the first value that gives x as an integer. k = 6 gives x = 1
Find y using the equation y = 9 - x = 8
The number we are looking for is 18 and the answer is D. It is divisible by 3 and the sum of its digits is equal to 9 and it is the smallest and positive whole number with such properties.
What is the smallest integer that is multiple of 5, 7 and 20? - a)70
- b)35
- c)200
- d)280
- e)140
Correct answer is option 'E'. Can you explain this answer?
What is the smallest integer that is multiple of 5, 7 and 20?
a)
70
b)
35
c)
200
d)
280
e)
140
|
|
Niti Choudhury answered |
It is the lcm of 5, 7 and 20 which is 140.
The answer is E
If N = 1000x + 100y + 10z, where x, y, and z are different positive integers less than 4, the remainder when N is divided by 9 is- a)2
- b)4
- c)6
- d)8
Correct answer is option 'C'. Can you explain this answer?
If N = 1000x + 100y + 10z, where x, y, and z are different positive integers less than 4, the remainder when N is divided by 9 is
a)
2
b)
4
c)
6
d)
8
|
|
Pallavi Sharma answered |
**Solution:**
To find the remainder when N is divided by 9, we need to consider the divisibility rule for 9.
According to the divisibility rule for 9, a number is divisible by 9 if the sum of its digits is divisible by 9.
Let's consider the number N = 1000x + 100y + 10z.
**Step 1: Find the sum of the digits of N**
The sum of the digits of N can be calculated as:
sum = x + y + z
**Step 2: Find the remainder when sum is divided by 9**
To find the remainder when sum is divided by 9, we can use the fact that the remainder when a number is divided by 9 is the same as the remainder when the sum of its digits is divided by 9.
Therefore, we need to find the remainder when sum is divided by 9.
**Step 3: Find the possible values of x, y, and z**
From the given information, we know that x, y, and z are different positive integers less than 4.
The possible values of x, y, and z are:
x = 1, 2, 3
y = 1, 2, 3
z = 1, 2, 3
**Step 4: Find the possible values of sum**
Using the values of x, y, and z, we can find the possible values of sum.
For x = 1, y = 1, z = 1, sum = 1 + 1 + 1 = 3
For x = 1, y = 1, z = 2, sum = 1 + 1 + 2 = 4
For x = 1, y = 1, z = 3, sum = 1 + 1 + 3 = 5
For x = 1, y = 2, z = 1, sum = 1 + 2 + 1 = 4
For x = 1, y = 2, z = 2, sum = 1 + 2 + 2 = 5
For x = 1, y = 2, z = 3, sum = 1 + 2 + 3 = 6
For x = 1, y = 3, z = 1, sum = 1 + 3 + 1 = 5
For x = 1, y = 3, z = 2, sum = 1 + 3 + 2 = 6
For x = 1, y = 3, z = 3, sum = 1 + 3 + 3 = 7
For x = 2, y = 1, z = 1, sum = 2 + 1 + 1 = 4
For x = 2, y = 1, z = 2, sum = 2 + 1 + 2 = 5
For x = 2, y = 1, z = 3, sum = 2 + 1 + 3 = 6
For x = 2, y = 2, z = 1, sum = 2 + 2 + 1 = 5
For x =
To find the remainder when N is divided by 9, we need to consider the divisibility rule for 9.
According to the divisibility rule for 9, a number is divisible by 9 if the sum of its digits is divisible by 9.
Let's consider the number N = 1000x + 100y + 10z.
**Step 1: Find the sum of the digits of N**
The sum of the digits of N can be calculated as:
sum = x + y + z
**Step 2: Find the remainder when sum is divided by 9**
To find the remainder when sum is divided by 9, we can use the fact that the remainder when a number is divided by 9 is the same as the remainder when the sum of its digits is divided by 9.
Therefore, we need to find the remainder when sum is divided by 9.
**Step 3: Find the possible values of x, y, and z**
From the given information, we know that x, y, and z are different positive integers less than 4.
The possible values of x, y, and z are:
x = 1, 2, 3
y = 1, 2, 3
z = 1, 2, 3
**Step 4: Find the possible values of sum**
Using the values of x, y, and z, we can find the possible values of sum.
For x = 1, y = 1, z = 1, sum = 1 + 1 + 1 = 3
For x = 1, y = 1, z = 2, sum = 1 + 1 + 2 = 4
For x = 1, y = 1, z = 3, sum = 1 + 1 + 3 = 5
For x = 1, y = 2, z = 1, sum = 1 + 2 + 1 = 4
For x = 1, y = 2, z = 2, sum = 1 + 2 + 2 = 5
For x = 1, y = 2, z = 3, sum = 1 + 2 + 3 = 6
For x = 1, y = 3, z = 1, sum = 1 + 3 + 1 = 5
For x = 1, y = 3, z = 2, sum = 1 + 3 + 2 = 6
For x = 1, y = 3, z = 3, sum = 1 + 3 + 3 = 7
For x = 2, y = 1, z = 1, sum = 2 + 1 + 1 = 4
For x = 2, y = 1, z = 2, sum = 2 + 1 + 2 = 5
For x = 2, y = 1, z = 3, sum = 2 + 1 + 3 = 6
For x = 2, y = 2, z = 1, sum = 2 + 2 + 1 = 5
For x =
The remainder when the positive integer m is divided by 7 is x. The remainder when m is divided by 14 is x + 7. Which one of the following could m equal? - a)45
- b)53
- c)72
- d)85
Correct answer is option 'B'. Can you explain this answer?
The remainder when the positive integer m is divided by 7 is x. The remainder when m is divided by 14 is x + 7. Which one of the following could m equal?
a)
45
b)
53
c)
72
d)
85
|
|
Arka Basu answered |
Solution:
Given, the remainder when the positive integer m is divided by 7 is x, and the remainder when m is divided by 14 is x+7.
Let's use the Chinese Remainder Theorem to solve this problem.
According to the Chinese Remainder Theorem, if we have a set of congruences of the form:
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
⋮
x ≡ ak (mod mk)
where m1, m2, …, mk are pairwise coprime integers (i.e., they have no common factors other than 1), then there exists a unique solution x modulo M = m1m2⋯mk.
We can apply this theorem to our problem as follows:
Let a1 = x, m1 = 7 (since the remainder when m is divided by 7 is x)
Let a2 = x+7, m2 = 14 (since the remainder when m is divided by 14 is x+7)
Note that m1 and m2 are not coprime, since they have a common factor of 7. However, we can still use the Chinese Remainder Theorem by dividing both sides of the second congruence by 7, which gives:
x ≡ (x+7) (mod 14)
x ≡ x+7 (mod 14)
0 ≡ 7 (mod 14)
This last congruence is true for all integers x, since 7 is a multiple of 14. Therefore, we can ignore it and focus on the first two congruences:
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
Substituting the values we have for a1, m1, a2, and m2, we get:
x ≡ x (mod 7)
x ≡ x+7 (mod 14)
To find the unique solution modulo M = m1m2 = 7×14 = 98, we can use the following formula:
x ≡ a1M2y1 + a2M1y2 (mod M)
where M1 = M/m1 = 14, M2 = M/m2 = 7, y1 and y2 are integers such that M1y1 ≡ 1 (mod m1) and M2y2 ≡ 1 (mod m2).
To find y1 and y2, we can use the Extended Euclidean Algorithm:
14y1 ≡ 1 (mod 7)
y1 ≡ 1 (mod 7)
y1 = 1
7y2 ≡ 1 (mod 14)
7y2 ≡ 15 (mod 14) (since 15 is also a solution)
y2 ≡ 2 (mod 14/7)
y2 ≡ 2 (mod 2)
y2 = 2
Substituting these values into the formula, we get:
x ≡ x×982 + (x+7)×1472 (mod 98)
x ≡ 53 (mod 98)
Therefore, the possible values for m are those that leave a remainder of 53 when divided by 98. Option B, 53, is the only one that satisfies this condition
Given, the remainder when the positive integer m is divided by 7 is x, and the remainder when m is divided by 14 is x+7.
Let's use the Chinese Remainder Theorem to solve this problem.
According to the Chinese Remainder Theorem, if we have a set of congruences of the form:
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
⋮
x ≡ ak (mod mk)
where m1, m2, …, mk are pairwise coprime integers (i.e., they have no common factors other than 1), then there exists a unique solution x modulo M = m1m2⋯mk.
We can apply this theorem to our problem as follows:
Let a1 = x, m1 = 7 (since the remainder when m is divided by 7 is x)
Let a2 = x+7, m2 = 14 (since the remainder when m is divided by 14 is x+7)
Note that m1 and m2 are not coprime, since they have a common factor of 7. However, we can still use the Chinese Remainder Theorem by dividing both sides of the second congruence by 7, which gives:
x ≡ (x+7) (mod 14)
x ≡ x+7 (mod 14)
0 ≡ 7 (mod 14)
This last congruence is true for all integers x, since 7 is a multiple of 14. Therefore, we can ignore it and focus on the first two congruences:
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
Substituting the values we have for a1, m1, a2, and m2, we get:
x ≡ x (mod 7)
x ≡ x+7 (mod 14)
To find the unique solution modulo M = m1m2 = 7×14 = 98, we can use the following formula:
x ≡ a1M2y1 + a2M1y2 (mod M)
where M1 = M/m1 = 14, M2 = M/m2 = 7, y1 and y2 are integers such that M1y1 ≡ 1 (mod m1) and M2y2 ≡ 1 (mod m2).
To find y1 and y2, we can use the Extended Euclidean Algorithm:
14y1 ≡ 1 (mod 7)
y1 ≡ 1 (mod 7)
y1 = 1
7y2 ≡ 1 (mod 14)
7y2 ≡ 15 (mod 14) (since 15 is also a solution)
y2 ≡ 2 (mod 14/7)
y2 ≡ 2 (mod 2)
y2 = 2
Substituting these values into the formula, we get:
x ≡ x×982 + (x+7)×1472 (mod 98)
x ≡ 53 (mod 98)
Therefore, the possible values for m are those that leave a remainder of 53 when divided by 98. Option B, 53, is the only one that satisfies this condition
For positive integers m and n, when m is divided by n the remainder is 4. Which of the following CANNOT be the value of n? - a)3
- b)5
- c)7
- d)12
Correct answer is option 'A'. Can you explain this answer?
For positive integers m and n, when m is divided by n the remainder is 4. Which of the following CANNOT be the value of n?
a)
3
b)
5
c)
7
d)
12
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Tejas Gupta answered |
Given:
Positive integers m and n such that when m is divided by n, the remainder is 4.
To find:
Which of the following values cannot be the value of n.
Solution:
Let's consider the division of m by n:
m = qn + r
where q is the quotient and r is the remainder. Given that the remainder is 4, we can rewrite the equation as:
m = qn + 4
This implies that m is 4 more than a multiple of n. We can rewrite this as:
m - 4 = qn
From this equation, we can observe that m - 4 must be a multiple of n.
Checking the answer choices:
a) 3: If n = 3, then m - 4 must be a multiple of 3. However, any positive integer minus 4 is not a multiple of 3. Therefore, 3 cannot be the value of n.
b) 5: If n = 5, then m - 4 must be a multiple of 5. This is possible, as there are multiples of 5 that are 4 more than another multiple of 5.
c) 7: If n = 7, then m - 4 must be a multiple of 7. This is possible, as there are multiples of 7 that are 4 more than another multiple of 7.
d) 12: If n = 12, then m - 4 must be a multiple of 12. This is possible, as there are multiples of 12 that are 4 more than another multiple of 12.
Therefore, the value that cannot be the value of n is 3 (option A).
Positive integers m and n such that when m is divided by n, the remainder is 4.
To find:
Which of the following values cannot be the value of n.
Solution:
Let's consider the division of m by n:
m = qn + r
where q is the quotient and r is the remainder. Given that the remainder is 4, we can rewrite the equation as:
m = qn + 4
This implies that m is 4 more than a multiple of n. We can rewrite this as:
m - 4 = qn
From this equation, we can observe that m - 4 must be a multiple of n.
Checking the answer choices:
a) 3: If n = 3, then m - 4 must be a multiple of 3. However, any positive integer minus 4 is not a multiple of 3. Therefore, 3 cannot be the value of n.
b) 5: If n = 5, then m - 4 must be a multiple of 5. This is possible, as there are multiples of 5 that are 4 more than another multiple of 5.
c) 7: If n = 7, then m - 4 must be a multiple of 7. This is possible, as there are multiples of 7 that are 4 more than another multiple of 7.
d) 12: If n = 12, then m - 4 must be a multiple of 12. This is possible, as there are multiples of 12 that are 4 more than another multiple of 12.
Therefore, the value that cannot be the value of n is 3 (option A).
If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?(1) When (z-3)2 is divided by 8, the remainder is 4(2) When 2z is divided by 8, the remainder is 2- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?
If z is a positive integer and r is the remainder when z2 + 2z + 4 is divided by 8, what is the value of r?
(1) When (z-3)2 is divided by 8, the remainder is 4
(2) When 2z is divided by 8, the remainder is 2
a)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
b)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
c)
BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
d)
EACH statement ALONE is sufficient to answer the question asked.
e)
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
|
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Devansh Chawla answered |
Steps 1 & 2: Understand Question and Draw Inferences
Given:
- Positive integer z
- z2+2z+4=8m+r,
- where m is an integer and r is the remainder
To find: The value of r
Step 3: Analyze Statement 1 independently
- When (z-3)2 is divided by 8, the remainder is 4
- So, we can write: (z−3)2=8k+4 , where k is an integer
- That is, z2–6z+9=8k+4
- Let us try to re-express the above equation in terms of the expression in the question:
- ? z2+2z+4
- z2 – 6z +( 4+5) = 8k + 4
- So, z2–6z+4=8k−1
- Adding 8z to both sides:
- z2 + 2z + 4 = 8 (k + z) - 1
- Now the expression on the left hand side of the equation is exactly the same as the expression in the question. On the right hand side, we see that the term 8(k+z) will be divisible by 8. But the remainder cannot be -1 because remainder cannot be negative. So, let’s rewrite the above equation:
- z2 + 2z + 4 = 8(k + z) - 8 + 7
- z2 + 2z + 4 = 8(k + z - 1) + 7
- The equation now is exactly comparable to the form 8m + r
- So, the remainder r = 7
Thus, Statement 1 alone is sufficient to answer the question.
Step 4: Analyze Statement 2 independently
- When 2z is divided by 8, the remainder is 2
- So we can write, 2z = 8n + 2, where n is some integer
- That is, z = 4n + 1
- In the above equation, the term 16n(n+1)
- will be divisible by 8
- So, the remainder when z2+2z+4 is divided by 8 will be 7
Thus, Statement 2 alone is sufficient to find a unique value of the remainder
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this step is not required
Answer: Option D
What is the remainder when 33 is divided by the integer y?
1) 90 < y < 100 ?
2) y is a prime number ?- a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'A'. Can you explain this answer?
What is the remainder when 33 is divided by the integer y?
1) 90 < y < 100 ?
2) y is a prime number ?
1) 90 < y < 100 ?
2) y is a prime number ?
a)
Exactly one of the statements can answer the question
b)
Both statements are required to answer the question
c)
Each statement can answer the question individually
d)
More information is required as the information provided is insufficient to answer the question
|
|
Kiran Chauhan answered |
Insufficient information. We need more equations to solve for y.
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Quantitative Reasoning for GRE
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