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When the positive integer A is divided by 5 and 7 [#permalink]
06 Jan 2012, 11:09

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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]
04 Aug 2014, 07:01

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Re: When the positive integer A is divided by 5 and 7 [#permalink]
04 Aug 2014, 20:27

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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