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