TITLE: Proof that $n+k+3$ divides $n(n+1)(n+2)(n+3) - k(k+1)(k+2)(k+3)$.
QUESTION [7 upvotes]: I'm looking for proof that
$$
(n+k+3) \mid n(n+1)(n+2)(n+3) - k(k+1)(k+2)(k+3)\\
n,k \in \mathbb N^*, n>k
$$
I tried using induction, but i'm not sure how it would work with 2 parameters.

REPLY [3 votes]: If $P$ is a polynomial with integer coefficients, then $$a-b \mid P(a)-P(b)$$ for integers $a$ and $b$. The desired conclusion follows by applying this with $a=n$, $b=-k-3$ and $$P(x) = x(x+1)(x+2)(x+3).$$ Indeed, we have $P(n)=n(n+1)(n+2)(n+3)$ and $P(-k-3) = (-k-3)(-k-2)(-k-1)(-k) = k(k+1)(k+2)(k+3)$.