# Math Help - number theory: proof required!

1. ## 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!)

Any nudges in the right direction? please?!

2. after two days of banging head against desk; I've come up with an unelegant solution

3. Notice that the last digit of two multiplied numbers depends only on the last digit of each number being multiplied.

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