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# When the positive integer A is divided by 5 and 7

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When the positive integer A is divided by 5 and 7 [#permalink]

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06 Jan 2012, 12:09
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When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainder is 3 and 4, respectively. Which of the following is a factor of A-B?

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30
[Reveal] Spoiler: OA

Last edited by Bunuel on 06 Jun 2013, 06:22, edited 3 times in total.
Edited the question
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13 Jan 2012, 06:55
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When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainder is 3 and 4, respectively. Which of the following is a factor of A-B?

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30

When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively: $$A=5q+3$$ (A could be 3, 8, 13, 18, 23, ...) and $$A=7p+4$$ (A 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 $$35$$.

Remainder will be the first common integer in above two patterns, hence $$18$$ --> so, to satisfy both this conditions A must be of a type $$A=35m+18$$ (18, 53, 88, ...);

The same for B (as the same info is given about B): $$B=35n+18$$;

$$A-B=(35m+18)-(35n+18)=35(m-n)$$ --> thus A-B must be a multiple of 35.

manhattan-remainder-problem-93752.html?hilit=derive#p721341
good-problem-90442.html?hilit=derive#p722552

Hope it helps.
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Re: When the positive integer A is divided by 5 and 7 [#permalink]

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04 Aug 2014, 08:01
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Re: When the positive integer A is divided by 5 and 7 [#permalink]

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04 Aug 2014, 21:27
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Although there are methods.. this particular problem does not require any methods..

A divided by 5,7 gives 3 and 4
B divided by 5,7 gives 3 and 4.

means A and B are equidistant from the multiples of 5,7. the LCM of 5 and 7 is 35, so numbers like A & B repeat after every 35 numbers. so 35 is the answer.
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Re: When the positive integer A is divided by 5 and 7 [#permalink]

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06 Aug 2014, 03:05
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With the remainders remaining same for both the required integers, it means there difference should be divisible by both 5 & 7

Out of all the five given options, only 35 stands out

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Re: When the positive integer A is divided by 5 and 7 [#permalink]

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13 Dec 2015, 21:39
Hello from the GMAT Club BumpBot!

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Re: When the positive integer A is divided by 5 and 7 [#permalink]

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14 Dec 2015, 00:40
Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.

When the positive integer A is divided by 5 and 7, the remainder is 3 and 4, respectively. When the positive integer B is divided by 5 and 7, the remainder is 3 and 4, respectively. Which of the following is a factor of A-B?

(A) 12
(B) 24
(C) 35
(D) 16
(E) 30

By the conditions we may put A=5K+3 = 7L+4, B=5M+3 = 7N+4. So we have A-B= 5K+3-5M-3 = 5*(K-M)= 7L+4 - 7N-4=7*(L-N).

We may conclude that A-B is divisible by 5 and 7. That means 35 is a factor of A-B.

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Re: When the positive integer A is divided by 5 and 7   [#permalink] 14 Dec 2015, 00:40
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