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15 Jun 2021, 05:30
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C
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:
47% (02:54) correct
53%
(03:21)
wrong
based on 62
sessions
History
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Time
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Not Attempted Yet
A and B when divided by 56 leave remainders of 48 and 32 respectively. When divided by 44, both leave a remainder of 24. If the sum of A and B is divisible by the sum of the divisors, then find the minimum value of (A + B).
(A) 1,200
(B) 1,800
(C) 2,600
(D) 3,400
(E) 3,800
(A) 1,200
(B) 1,800
(C) 2,600
(D) 3,400
(E) 3,800
15 Jun 2021, 17:51
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Bunuel
If we focus on A + B, we know A + B when divided by 44 must leave a remainder of \(24 + 24\) -> \(48\) -> \(4\) since we can divide out that extra 44. If we divide A + B by 56 then the remainder must be \(48 + 32\) -> \(80\) -> \(24\). So our goal is to find a number that can be satisfy both forms, \(44*i + 4\) and \(56*j + 24\), where i and j are integers.
Equate these and we can simplify this down to \(11*i + 1 = 14*j + 6\). In an attempt to find a chain of integer solutions, try adding 14's onto 6 to find a multiple of 11 plus 1 since multiples of 11 are easier to count. If we scroll through the numbers we have 6, 20, 34, and bingo stop there since 34 is a multiple of 11 plus 1. Then (i, j) = (3, 2) is a viable solution. And by adding 14's to the i and 11's to the j values, we will get additional (and infinitely many) solutions.
Now let's go back to the original number, the first satisfying number would be \(44*3 + 4 = 136\) from our i = 3 case. We can get more solutions by adding 14 to the i values or 11 to the j values, hence 56*11 = 560 + 56 = 616 increments. We want a number that is divisible by 56 + 44 = 100, start with 136 and check how many 16's we need to add to reach a whole hundred. We are missing 64 = 4 * 16, so add four more of 616's.
Finally the answer would be \(616*4 + 136 = 2464 + 136 = 2600.\)
Ans: C
General Discussion
21 May 2026, 05:56
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Bunuel
A = 44a + 24
B = 44b + 24
A + B = 44(a + b) + 48 = Multiple of 44 + 4
So the options are 4 more than a multiple of 44 i.e a multiple of 11. Check them.
(A) 1,200
1196 is not divisible by 11. Ignore.
(B) 1,800
1796 is not divisible by 11. Ignore.
(C) 2,600
2596 is divisible by 11 (diff between sum of odd and even digits is 0)
(D) 3,400
3396 is not divisible by 11. Ignore.
(E) 3,800
3796 is not divisible by 11. Ignore.
Answer (C)
Division discussed here: https://youtu.be/A5abKfUBFSc
