number theory: proof required!
I've been given the following problem:
Prove that the last digit of the product of two consecutive integers is 0, 2 or 6.
So far, I can prove it's not an odd number:
n and (n+1) are two consecutive integers - one is odd, and the other even, so, let n=2x and (N+1) = 2x + 1
so n(n+1) = 2x(2x+1) and so an even number.
and write out a table:
0x1 = 0
1x2 = 2
2x3 = 6
3x4 = 12 (last digit is 2)
4x5 = 20 (last digit is 0)
5x6 = 30 (last digit is 0)
6x7 = 42 (last digit is 2)
7x8 = 56 (last digit is 6)
8x9 = 72 (last digit is 2)
and I can justify to myself that clearly when you multiply any two consecutive integers, the last digit will be a 0, 2 or 6; but I can't prove this (to either myself or my uni lecturer!)
Any nudges in the right direction? please?! (Headbang)