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All questions of Remainders for UPSC CSE Exam
Find the Remainder77/24- a)3
- b)5
- c)7
- d)1
Correct answer is option 'C'. Can you explain this answer?
Find the Remainder
77/24
a)
3
b)
5
c)
7
d)
1
|
Palak Patel answered |
**Explanation:**
To find the remainder when 77 is divided by 24, we can use the division algorithm which states that any integer a can be expressed as a = bq + r, where b is the divisor, q is the quotient, and r is the remainder. In this case, a = 77, b = 24, q is the quotient we need to find, and r is the remainder we are looking for.
**Step 1: Divide the Dividend by the Divisor**
Dividing 77 by 24, we get:
77 ÷ 24 = 3 remainder 5
This means that when 77 is divided by 24, we get a quotient of 3 and a remainder of 5.
**Step 2: Verify the Remainder**
To confirm that the remainder is indeed 5, we can multiply the quotient by the divisor and add the remainder:
3 × 24 + 5 = 72 + 5 = 77
Since the result is equal to the dividend 77, we can conclude that the remainder is indeed 5 when 77 is divided by 24.
Therefore, the correct answer is option C) 5.
To find the remainder when 77 is divided by 24, we can use the division algorithm which states that any integer a can be expressed as a = bq + r, where b is the divisor, q is the quotient, and r is the remainder. In this case, a = 77, b = 24, q is the quotient we need to find, and r is the remainder we are looking for.
**Step 1: Divide the Dividend by the Divisor**
Dividing 77 by 24, we get:
77 ÷ 24 = 3 remainder 5
This means that when 77 is divided by 24, we get a quotient of 3 and a remainder of 5.
**Step 2: Verify the Remainder**
To confirm that the remainder is indeed 5, we can multiply the quotient by the divisor and add the remainder:
3 × 24 + 5 = 72 + 5 = 77
Since the result is equal to the dividend 77, we can conclude that the remainder is indeed 5 when 77 is divided by 24.
Therefore, the correct answer is option C) 5.
What is the remainder when [(919) + 6] is divided by 8- a)6
- b)7
- c)0
- d)3
Correct answer is option 'B'. Can you explain this answer?
What is the remainder when [(919) + 6] is divided by 8
a)
6
b)
7
c)
0
d)
3
|
Kiran Sarkar answered |
The given expression is in the form (x19) + c; where ‘c’ is a constant ……….(a polynomial in x).
Now 8 = 9 1; means a polynomial in the form of x 1
So according to the remainder theorem when a polynomial is divided by another of the form x 1, the remainder is equal to p(1) where p is the polynomial itself. So using remainder theorem, the remainder is (119) + 6 = 1 + 6 = 7 (option ‘B’)
Find smallest number that leaves remainder 3, 4, 5 when divided by 5, 6, 7 respectively and leaves remainder 1 when divided by 11.- a)208
- b)426
- c)600
- d)628
Correct answer is option 'D'. Can you explain this answer?
Find smallest number that leaves remainder 3, 4, 5 when divided by 5, 6, 7 respectively and leaves remainder 1 when divided by 11.
a)
208
b)
426
c)
600
d)
628
|
|
Aditi Roy answered |
We have just seen above in TYPE-2 how to tackle the first part of the questionThus the number for the first part would be the [(LCM of 5, 6, 7) – (Common difference of divisors and their remainders)] i.e. 210 – 2 = 208
Here now, we have one more condition to satisfy i.e. remainder 1 when divided by 11
Here we should remember that if LCM of the divisors is added to a number; the corresponding remainders do not change i.e if we keep adding 210 to 208… the first 3 conditions for remainders will continue to be fulfilled.
Therefore now, let 208 + 210k be the number that will satisfy the 4th condition i.e. remainder 1 when (208 + 210k)/11
Now let’s see how
The expression (208 + 210k)/11 = 208/11 + 210k/11
Now the remainder when 208 is divided by 11 = 10
And remainder when 210k is divided by 11 = 1*k = k
The expression (208 + 210k)/11 = 208/11 + 210k/11
Now the remainder when 208 is divided by 11 = 10
And remainder when 210k is divided by 11 = 1*k = k
Therefore the sum of both the remainders i.e. 10 + k should leave remainder 1 on division of the number by 11
Obviously k = 2
Hence the number = 208 + 210*2 = 628 (option ‘D’)
Obviously k = 2
Hence the number = 208 + 210*2 = 628 (option ‘D’)
Numbers 11284 and 7655, when divided by a certain number of three digits, leave the same remainder. Find that number of 3 digits and their sum.- a)191 & 11
- b)911 & 11
- c)181 & 10
- d)811 & 10
Correct answer is option 'A'. Can you explain this answer?
Numbers 11284 and 7655, when divided by a certain number of three digits, leave the same remainder. Find that number of 3 digits and their sum.
a)
191 & 11
b)
911 & 11
c)
181 & 10
d)
811 & 10
|
Avik Majumdar answered |
One has to remember that each factor of the difference of two numbers gives the same remainder if those numbers are divided by it.
Now the difference here = 11284 – 7655 = 3629
Factors of 3629 are 1, 19, 191 and 3629
But we have to find the three digit number here, so 191 is the required number and their sum is 1+9+1 = 11 (option ‘A’)
Factors of 3629 are 1, 19, 191 and 3629
But we have to find the three digit number here, so 191 is the required number and their sum is 1+9+1 = 11 (option ‘A’)
What is the remainder when [7(4n + 3)]*6n is divided by 10; where ‘n’ is a positive integer.- a)2
- b)4
- c)8
- d)6
Correct answer is option 'C'. Can you explain this answer?
What is the remainder when [7(4n + 3)]*6n is divided by 10; where ‘n’ is a positive integer.
a)
2
b)
4
c)
8
d)
6
|
|
Rishika Menon answered |
[7(4n + 3)]6n
= 74n x 73 x 6n
= 492n x 73 x 6n
= When each factor is divided by 10 the remainders in each case = (-1)2n, 3, and 6 (6 when raised to the power of any natural number is divided by 10 always gives remainder as 6 itself)
= 74n x 73 x 6n
= 492n x 73 x 6n
= When each factor is divided by 10 the remainders in each case = (-1)2n, 3, and 6 (6 when raised to the power of any natural number is divided by 10 always gives remainder as 6 itself)
So, all the remainders thus found above are 1, 3 and 6
So their multiplication= 1*3*6= 18
So the remainder after 18 has been divided by 10 = 8 (option ‘C’)
So their multiplication= 1*3*6= 18
So the remainder after 18 has been divided by 10 = 8 (option ‘C’)
2525 is divided by 26, the remainder is?
- a)1
- b)2
- c)24
- d)25
Correct answer is option 'D'. Can you explain this answer?
2525 is divided by 26, the remainder is?
a)
1
b)
2
c)
24
d)
25
|
|
Preethi Roy answered |
(xn + an) is divisible by (x + a) when n is odd
∴ (2525 + 125) is divisible by (25 + 1)
⇒ (2525 + 1) is divisible by 26
⇒ On dividing 2525 by 26, we get (26 - 1) = 25 as remainder
∴ (2525 + 125) is divisible by (25 + 1)
⇒ (2525 + 1) is divisible by 26
⇒ On dividing 2525 by 26, we get (26 - 1) = 25 as remainder
Find the remainders in
211/5- a)0
- b)1
- c)2
- d)3
Correct answer is option 'D'. Can you explain this answer?
Find the remainders in
211/5
211/5
a)
0
b)
1
c)
2
d)
3
|
|
Nishanth Desai answered |
1. 211/5
In questions like this we should avoid using the remainder theorem as it can really be difficult when the power of a number (greater than 1) which is derived from the remainder theorem is so high. Better convert the base in powers of such numbers which are easily divisible by the divisor, like:
In questions like this we should avoid using the remainder theorem as it can really be difficult when the power of a number (greater than 1) which is derived from the remainder theorem is so high. Better convert the base in powers of such numbers which are easily divisible by the divisor, like:
211/5 = 24 x 24 x 23= 16 x 16 x 8
On dividing 16 by 5 we get 1 as the remainder; and if 8 is divided by 5 we get 3
So the multiplication of all the remainders
= 1 x 1 x 3 = 3 which is our answer (option ‘A’)
So the multiplication of all the remainders
= 1 x 1 x 3 = 3 which is our answer (option ‘A’)
64329 is divided by a certain number. While dividing, the numbers 175, 114 and 213 appear as three successive remainders, the divisor is?- a)184
- b)224
- c)234
- d)6250
Correct answer is option 'C'. Can you explain this answer?
64329 is divided by a certain number. While dividing, the numbers 175, 114 and 213 appear as three successive remainders, the divisor is?
a)
184
b)
224
c)
234
d)
6250
|
|
Nishanth Desai answered |
We have three remainders, means the number comprising of the first digits i.e. 643 was divided first and we got 175 as the remainder.
Now according to DIVIDEND = DIVISOR x QUOTIENT + REMAINDER
=> DIVISOR x QUOTIENT = DIVIDEND – REMAINDER
=> DIVISOR x QUOTIENT = 643 – 175 = 468
=> DIVISOR x QUOTIENT = DIVIDEND – REMAINDER
=> DIVISOR x QUOTIENT = 643 – 175 = 468
We see that 468 is divisible by 234 only among all the answer options; so 234 (option ‘C’) is the divisor we need.
Find smallest number that leaves remainder 3, 5, 7 when divided by 4, 6, 8 respectively.- a)23
- b)25
- c)35
- d)37
Correct answer is option 'A'. Can you explain this answer?
Find smallest number that leaves remainder 3, 5, 7 when divided by 4, 6, 8 respectively.
a)
23
b)
25
c)
35
d)
37
|
|
Charvi Banerjee answered |
See carefully the difference between the divisor and remainder is having a certain trend. i.e.4 – 3 = 6 – 5 = 8 – 7=1
In such questions, take LCM of divisors and subtract the common difference from it.Now the LCM of 4, 6, 8 = 24Therefore the required number here = 24 – 1 = 23 (option ‘A’)
When (6767 + 67) is divided by 68, the remainder is?- a)1
- b)63
- c)66
- d)67
Correct answer is option 'C'. Can you explain this answer?
When (6767 + 67) is divided by 68, the remainder is?
a)
1
b)
63
c)
66
d)
67
|
|
Raksha Deshpande answered |
The given expression is in the form x67 + x ……….(a polynomial in x)
Now 68 = 67 + 1; means x +1
Now 68 = 67 + 1; means x +1
So according to the remainder theorem when a polynomial is divided by another of the form x + 1, the remainder is equal to p(1) where p is the polynomial itself.
So the remainder is 167 + (1) = 1 + (1) = 2
So the remainder is 167 + (1) = 1 + (1) = 2
But the remainder should not be described negative of a number; in such a situation it is added to the divisor to find the actual.
So the remainder is 2 + 68 = 66 (option ‘C’)
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