A fifth degree polynomial $P(x)$ with integral coefficients. A fifth degree polynomial $P(x)$ with integral coefficients takes on values $0,1,2,3,4$ at $x=0,1,2,3,4$, respectively.
Which of the following is a possible value for $P(5)$?
A) $5$ 
B) $24$ 
C) $125$ 
D)None of the above

Any polynomial of degree at most five, fulfilling the given interpolation conditions, has the form
\[
  P(x) = x + a\prod_{i=0}^4 (x-i)
\]
for some $a \in \mathbb Z$ (we can see this if we write $P$ in the Newton basis, for example). So $P(5) = 5 + 5!a = 5 + 120a$. Now A) corresponds to $a = 0$, B) to $a = \frac{19}{120}$ and C) to $a=1$. As we want $P$ to be of fifth degree, we need $a \ne 0$, and as $\frac{19}{120} \not\in\mathbb Z$, the correct answer is C), giving $P(x) = x + \prod_{i=0}^4(x-i)$.
