PRMO 2017 Sample Qn

We use the fact that for a polynomial $P(x)$ with integer coefficients, $a-b\mid(P(a)-P(b)\\$ So, for the polynomial in the question: $n\mid P(n)-P(0)\\$ $n\mid P(0)\\$ Since this is true for all $n\in\mathbb{N}$ , every positive integer divides $P(0)$ . Thus, the only value it can take is $\boxed{0}$

given P(n) is divisible by n for each +ve value of n. therefore the polynomial P(n) has no constant. Thus p(0) = 0