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26 Jul 2026, 09:49
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
12% (02:00) correct
88%
(02:21)
wrong
based on 26
sessions
History
Date
Time
Result
Not Attempted Yet
755, 811, 832
When each of the numbers above is divided by a certain positive integer, the remainder is the positive integer r. What is the remainder when their sum is divided by r?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
When each of the numbers above is divided by a certain positive integer, the remainder is the positive integer r. What is the remainder when their sum is divided by r?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
27 Jul 2026, 20:53
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\(811-755=56,\quad 832-811=21\)
Since all three numbers leave the same remainder when divided by the unknown divisor, the divisor must divide both 56 and 21, so it divides\( \gcd(56,21)=7\). Since the remainder is positive, the divisor cannot be 1. Hence the divisor is 7.
Using just one number to find r, \(755=7\cdot107+6\) so r=6.
Also, 755+811+832=2398=2400-2.
Since 2400 is divisible by 6, the remainder when 2398 is divided by 6 is 6-2=4.
Answer: (E)
Since all three numbers leave the same remainder when divided by the unknown divisor, the divisor must divide both 56 and 21, so it divides\( \gcd(56,21)=7\). Since the remainder is positive, the divisor cannot be 1. Hence the divisor is 7.
Using just one number to find r, \(755=7\cdot107+6\) so r=6.
Also, 755+811+832=2398=2400-2.
Since 2400 is divisible by 6, the remainder when 2398 is divided by 6 is 6-2=4.
Answer: (E)
27 Jul 2026, 21:53
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Oh this one is nasty, love it. The 0% correct rate on this thread checks out, because the way most people approach remainder problems here almost guarantees a wrong turn somewhere.
1. Call the mystery divisor n and the common remainder r. Since 755, 811, and 832 all leave the same remainder r when divided by n, the pairwise differences between them have to be clean multiples of n.
2. So n divides 811-755=56, n divides 832-811=21, and n divides 832-755=77.
3. Take the GCD of 56, 21, and 77. GCD of 56 and 21 is 7, and GCD of 7 and 77 is still 7. So n is a factor of 7, meaning n is either 1 or 7.
4. Here's the trap. n=1 looks fine on paper but it can't work, because dividing by 1 always leaves remainder 0, and the problem tells you r is a positive integer. So n=1 is out, n=7 is the answer.
5. Check it: 755 divided by 7 is 107 remainder 6, 811 divided by 7 is 115 remainder 6, 832 divided by 7 is 118 remainder 6. Consistent, so r=6.
6. Now the part everyone forgets, read the question again. It asks for the remainder when the sum is divided by r, not by n. Sum is 755+811+832=2398. Divide that by r=6, and you get 399 remainder 4.
Answer is E. I think nobody's gotten this right yet because people either forget to rule out n=1, or they get to n=7 and just divide the sum by 7 out of habit instead of by r. In this problem those two paths happen to land on the same remainder, which is almost worse because it teaches you the wrong lesson for next time. Moral of the story: on remainder questions, slow down on what exactly you're dividing by before you touch an answer choice.
1. Call the mystery divisor n and the common remainder r. Since 755, 811, and 832 all leave the same remainder r when divided by n, the pairwise differences between them have to be clean multiples of n.
2. So n divides 811-755=56, n divides 832-811=21, and n divides 832-755=77.
3. Take the GCD of 56, 21, and 77. GCD of 56 and 21 is 7, and GCD of 7 and 77 is still 7. So n is a factor of 7, meaning n is either 1 or 7.
4. Here's the trap. n=1 looks fine on paper but it can't work, because dividing by 1 always leaves remainder 0, and the problem tells you r is a positive integer. So n=1 is out, n=7 is the answer.
5. Check it: 755 divided by 7 is 107 remainder 6, 811 divided by 7 is 115 remainder 6, 832 divided by 7 is 118 remainder 6. Consistent, so r=6.
6. Now the part everyone forgets, read the question again. It asks for the remainder when the sum is divided by r, not by n. Sum is 755+811+832=2398. Divide that by r=6, and you get 399 remainder 4.
Answer is E. I think nobody's gotten this right yet because people either forget to rule out n=1, or they get to n=7 and just divide the sum by 7 out of habit instead of by r. In this problem those two paths happen to land on the same remainder, which is almost worse because it teaches you the wrong lesson for next time. Moral of the story: on remainder questions, slow down on what exactly you're dividing by before you touch an answer choice.
