number theory: proof required!

• February 1st 2011, 06:15 AM
Whiskyfied
number theory: proof required!
hi

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!)

Thus you just need to verify it for $n=0$ through $n=9$.