I stumbled across the following problem and found it cute.
Problem: We are given that 19 divides 23028, 31882, 86469, 6327, and 61902. Show that 19 divides the following determinant:
Multiply the first column by 104, the second by 103, third by 102 and fourth by 10 – this will scale the value of the determinant by 104+3+2+1=1010, which is coprime to 19. Now add the last four columns to the first one – this will not change the value of the determinant. Finally notice the first column now reads 23028,31882,86469,6327, and 61902: each is a multiple of 19 so we can factor a nineteen cleanly out the determinant.