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# When positive integer x is divided by 5, the remainder is 3; and when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3; and when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y? a)15b)20c)28d)35Correct answer is option 'D'. Can you explain this answer? Related Test: Test: Remainders- 1

## GMAT Question

 Aakash Pandey Jul 31, 2018
When the positive integer x is divided by 5 and 7, the remainder is 3 and 4, respectively: x=5q+3x=5q+3 (x could be 3, 8, 13, 18, 23, ...) and x=7p+4x=7p+4 (x 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 3535.

Remainder will be the first common integer in above two patterns, hence 1818 --> so, to satisfy both this conditions x must be of a type x=35m+18x=35m+18 (18, 53, 88, ...);

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

x−y=(35m+18)−(35n+18)=35(m−n)x−y=(35m+18)−(35n+18)=35(m−n) --> thus x-y must be a multiple of 35.