Pick four integers. Calculate their six pairwise differences. Multiply all those differences. Prove that the result is a multiple of 12.

Of the six differences, at least one is a multiple of 3.

Consider each of the original integers each of them must be congruent to 0, 1 or 2 mod 3 so at least two of them must be the same.  This means one difference is a multiple of 3.

Of the six differences, either one os a multiple of 4 or at least two are multiples of 2.

The orignal integers will each be congruent to 0, 1, 2 or 3 mod 4.  If two are the same they differ by a multiple of 4.  If all are different the pair congruent to 0 and 2 and the pair congruent to 1 and 3 differ by a multiple of 2.

These differences supply the factors sufficient for the product to be a multiple of 12.

Posted by Jer on 2007-03-28 07:33:45