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.

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.

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