# Number Theory divisibility proof

1. Mar 1, 2010

### jersiq1

1. The problem statement, all variables and given/known data
Prove that for any n $$\in$$ Z+, the integer (n(n+1)(n+2) + 21) is divisible by 3

2. Relevant equations

A previously proved lemma (see below)

3. The attempt at a solution

I sort of just need a nudge here. I have a previously proven lemma which states:

If d|a and d|b, then d|(a+b)

So armed with this I see that obviously 3|21 and all that remains is to prove n(n+1)(n+2) is divisible by 3. I have tried expanding, which didn't seem to help.

2. Mar 1, 2010

### VeeEight

n(n+1)(n+2) is the product of __ consecutive integers.

3. Mar 1, 2010

### jersiq1

Wow! staring me in the face. Thanks.

4. Mar 1, 2010

Cheers :)