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Test: Remainders- 1 - GMAT MCQ


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15 Questions MCQ Test - Test: Remainders- 1

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Test: Remainders- 1 - Question 1
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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?

Detailed Solution for Test: Remainders- 1 - Question 1
  • When the positive integer x is divided by 5 and 7, the remainder is 3 and 4, respectively: x=5q+3 (x could be 3, 8, 13, 18, 23, ...) and x=7p+4 (x could be 4, 11, 18, 25, ...).
  • There is a way to derive a general formula based on the above two statements:
  • Divisor will be the least common multiple of the above two divisors 5 and 7, hence 35.
  • The remainder will be the first common integer in the above two patterns, hence 18 --> so, to satisfy both these conditions x must be of a type x=35m+18 (18, 53, 88, ...);
  • The same for y (as the same info is given about y): y=35n+18
  • x−y=(35m+18)−(35n+18)=35(m−n)) --> thus x-y must be a multiple of 35.
     
Test: Remainders- 1 - Question 2
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If n is a positive integer, what is the remainder when 38n+3 + 2 is divided by 5? 

Detailed Solution for Test: Remainders- 1 - Question 2
  • The units digit of 3 in positive integer power has cyclicity of 4 for the unis digit:
  • 31 --> the units digit is 3;
  • 32 --> the units digit is 9;
  • 33 --> the units digit is 7;
  • 34 --> the units digit is 1;
  • 35 --> the units digit is 3 AGAIN;
  • So, the units digit repeats the following pattern {3-9-7-1}-{3-9-7-1}-.... 38n+3 will have the same units digit as 33, which is 7 (remainder when 8n+3 divided by cyclicity 4 is 3). Thus the last digit of 38n+3+2 will be 7+2=9. Any positive integer with the unis digit of 9 divided by 5 gives the remainder of 4.
     
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Test: Remainders- 1 - Question 3
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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 

Detailed Solution for Test: Remainders- 1 - Question 3
  • (1) n=(k+1)3
  • =k3+3k2+3k+1
  • =k(k2+3k+3)+1
  • --> first term, k(k2+3k+3) is obviously divisible by k and 1 divide by k yields the remainder of 1 (as k>1). Sufficient.
  • (2) k=5. Know nothing about n, hence insufficient.
  • Hence exactly one of the statements in the question is correct.
     
Test: Remainders- 1 - Question 4
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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. 
Detailed Solution for Test: Remainders- 1 - Question 4
  • St1: x= 6(m+3) where m is the quotient. But no info about y. Clearly insufficient
  • St2: y= 6(n+1) where n is the quotient . Insufficient
  • Combining,
  • x+y= 6(m+n+4)/6
  • Hence remainder will be 4. Sufficient


 

Test: Remainders- 1 - Question 5
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When 20 is divided by the positive integer k, the remainder is k – 2. Which of the following is a possible value of k? 
Detailed Solution for Test: Remainders- 1 - Question 5
  • Dividend=Divisor x Quotient + Remainder
  • 20 = k x q+k-2......k x q+k
  • =20+2=22
  • k(q+1)=22=2 x 11=1 x 22
  • Possible values of k are 2 and 11 or 22.
  • Hence, we have the option as 11 hence option D is correct.
     
Test: Remainders- 1 - Question 6
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Seven different numbers are selected from the integers 1 to 100, and each number is divided by 7. What is the sum of the remainders?

1)  The range of the seven remainders is 6. ?

2)  The seven numbers selected are consecutive integers. ?
Detailed Solution for Test: Remainders- 1 - Question 6
  • Correct Answer :- b
  • Explanation : The trick here is to know that remainder is always non-negative integer less than divisor 0≤r<d0≤r<d, so in our case 0≤r<70≤r<7.
  • So the remainder upon division of any integer by 7 can be: 0, 1, 2, 3, 4, 5, or 6 (7 values).
  • (1) The range of the seven remainders is 6 --> if we pick 6 different multiples of 7 (all remainders 0) and the 7th number 6 (remainder 6) then the range would be 6 and the sum also 6. But if we pick 7 consecutive integers then we'll have all possible remainders: 0, 1, 2, 3, 4, 5, and 6 and their sum will be 21. Not sufficient.
  • (2) The seven numbers selected are consecutive integers --> ANY 7 consecutive integers will give us all remainders possible: 0, 1, 2, 3, 4, 5, and 6. It does not matter what the starting integer will be: if it's say 11 then the remainder of 7 consecutive integers from 11 divided by 7 will be: 4, 5, 6, 0, 1, 2, and 3 and if starting number is say 14 then the remainder of 7 consecutive integers from 14 divided by 7 will be: 0, 1, 2, 3, 4, 5 and 6. So in any case sum=0+1+2+3+4+5+6=21. Sufficient.
Test: Remainders- 1 - Question 7
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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
Detailed Solution for Test: Remainders- 1 - Question 7
  • 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
     
Test: Remainders- 1 - Question 8
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If n is a positive integer, what is the remainder when (7(4n+3))(6n) is divided by 10? 
Detailed Solution for Test: Remainders- 1 - Question 8
  • last digits of 7^m
  • 71 : 7
  • 72 : 9
  • 73 : 3
  • 74 : 1
  • 75 : 7
  • period=4 ==> last digit: 7(4n+3) = 7(4+3) = 3
  • last digit of 6n always 6.
  • 3*6 ==> 8 - reminder (last digit)
Test: Remainders- 1 - Question 9
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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. 
Detailed Solution for Test: Remainders- 1 - Question 9

Test: Remainders- 1 - Question 10
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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. 
Detailed Solution for Test: Remainders- 1 - Question 10
  • (1) n is not divisible by 2. Insufficient on its own, but this statement says that n=odd--> n−1 and n+1 are consecutive even integers --> (n−1)(n+1) must be divisible by 8 (as both multiples are even and one of them will be divisible by 4. From consecutive even integers one is divisible by 4: (2, 4); (4, 6); (6, 8); (8, 10); (10, 12), ...).
  • (2) n is not divisible by 3. Insufficient on its own, but form this statement either n−1 or n+1 must be divisible by 3 (as n−1, n, and n+1 are consecutive integers, so one of them must be divisible by 3, we are told that it's not n, hence either n−1 or n+1).
  • (1)+(2) From (1) (n−1)(n+1) is divisible by 8, from (2) it's also divisible by 3, therefore it must be divisible by 8∗3=24, which means that remainder upon division (n−1)(n+1) by 24 will be 0. Sufficient.
     
Test: Remainders- 1 - Question 11
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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. 
Detailed Solution for Test: Remainders- 1 - Question 11
  • (1) The sum of the digits of x is 5 --> x can be 14, 41, 23, 32, or 50. Each of this numbers gives the remainder of 2 when divided by 3. Sufficient.
  • (2) The remainder when x is divided by 9 is 5 --> x=9q+5=9q+3+2=3(3q+1)+2--> the remainder when x is divided by 3 is 2. Sufficient.
     
Test: Remainders- 1 - Question 12
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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 ?
Detailed Solution for Test: Remainders- 1 - Question 12
  • In many remainders questions, it's enough just to find a couple of numbers that 'work' with the given information, and if you simply list the first few numbers that satisfy each statement, it's easy to judge if the statements together are sufficient:
  • 1) x = 5, 17, 29, 41, 53, 65, 77, 89 ...
  • 2) x = 11, 29, 47, 65, 83, 101 ...
  • Since 29 and 65 give different remainders when you divide by 8, the answer is E.
  • More abstractly, when combining two statements like the above, the pattern will be based on the LCM of the two divisors. Here, we can consider dividing x by 36 = LCM(12, 18).
  • Notice that, if Statement 1 is true, the remainder will be 5, 17 or 29 when x is divided by 36.
  • If Statement 2 is true, the remainder will be 11 or 29 when x is divided by 36.
  • If both Statements are true, the remainder therefore must be 29 when x is divided by 36.
  • That's still not sufficient, as above; x could be 29, or x could be 65.
  • Therefore, option D is correct.
Test: Remainders- 1 - Question 13
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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? 
Detailed Solution for Test: Remainders- 1 - Question 13
  • 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
Test: Remainders- 1 - Question 14
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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) ) ? 
Detailed Solution for Test: Remainders- 1 - Question 14
  • 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

 

Test: Remainders- 1 - Question 15
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How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3? 
Detailed Solution for Test: Remainders- 1 - Question 15
  • 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 of 1 when divided by 3
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