How do you find the value of $N$ given $P(N) = N+51$ and other information about the polynomial $P(x)$? Problem:
Let $P(x)$ be a polynomial with integer coefficients such that $P(21)=17$, $P(32)=-247$, $P(37)=33$. If $P(N) = N + 51$ for some positive integer $N$, then find $N$.

I can't think of anyway to begin this question so any help will be appreciated.

Since $P$ is with integer coefficients then 
$$\forall x,y \in \mathbb{Z}:(x-y) | \left(p(x) - p(y)\right)$$ then we will get 
$$p(n)-p(21) = n+34 \Rightarrow n-21 |n+34 \Rightarrow  n-21 | 55$$
$$p(n)-p(32) = n+298 \Rightarrow n-32 |n-298 \Rightarrow n+298 |330$$
$$p(n)-p(37) = n+18 \Rightarrow n-37 |n+18 \Rightarrow n-37 |55 $$
So I think if we do all calculations  we are done.
