1. ## Divisibility problem.

Prove n(n+4)(n+5) is divisible by 6.

2. ## Re: Divisibility problem.

Originally Posted by louis33
Prove n(n+4)(n+5) is divisible by 6.
Use induction. If $N(N+4)(N+5)$ is divisible by $6$ then it is plainly obvious so is $(N+1)(N+1+4)(N+1+5)$.

3. ## Re: Divisibility problem.

One of $N+4$ and $N+5$ is divisible by 2 (one is odd, the other even).
One of:
$N$
$N+4 = (N+1)+3$
$N+5 = (N+2)+3$
is divisible by 3 (the other two are not).

Any product that is divisible by both 2 and 3 is divisible by 6.

4. ## Re: Divisibility problem.

Originally Posted by louis33
Prove n(n+4)(n+5) is divisible by 6.
n must be an integer, of course...

5. ## Re: Divisibility problem.

Originally Posted by louis33
Prove n(n+4)(n+5) is divisible by 6.
$\displaystyle n(n + 4)(n + 5) =$

$\displaystyle n^3 + 9n^2 + 20n =$

$\displaystyle n^3 + 3n^2 + 2n + 6n^2 + 18n =$

$\displaystyle n(n + 1)(n + 2) + 6(n^2 + 3n)$