Integer Coefficients

We can use the fact that $a - b | P(a) - P(b)$ for any polynomial $P$ and any integers $a$ and $b$ .

This fact is simple to prove using the factorisation $x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1})$ .

Then let $a$ be such that $P(a) = 5$ .

Then $a$ must satisfy:

$(a - 1) | 4$

$(a - 2) | 3$

$(a - 3) | 2$

(where $a | b$ means b is divisible by $a$ )

The last condition means that $a = 4$ or $a = 5$ . However, $a = 4$ doesn't satisfy the other two conditions so $a = 5$ . This is clearly possible as a value since we can take the polynomial $P(x) = x$ .