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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?

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.

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.

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!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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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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