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Seventeen divides expression? (Posted on 2015-04-26) Difficulty: 3 of 5
x, y are integers, and 17 divides both the expressions x2-2xy+y2-5x+7y and x2-3xy+2y2+x-y.

Does 17 always divide xy-12x+15y, for every choice of x and y?

If not, give an example.
If so, prove it.

No Solution Yet Submitted by K Sengupta    
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solution | Comment 2 of 3 |
quick search of all combinations of remainders for x,y mod 17 gives that only when (x,y) mod 17 is (0,0), (3,2), or (5,3) is both of the given expressions divisible by 17.

For each of these 3 combinations, xy-12+15y is also divisible by 17.

Thus it is always true.

  Posted by Daniel on 2015-04-27 11:23:51
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