Question 12
(ITA) Let $p$ be a prime number and let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that:
(i) $f(0)=0$, $f(1)=1$;
(ii) for every positive integer $n$, the remainder of the division of $f(n)$ by $p$ is either $0$ or $1$.
Prove that $d\ge p-1$.
(i) $f(0)=0$, $f(1)=1$;
(ii) for every positive integer $n$, the remainder of the division of $f(n)$ by $p$ is either $0$ or $1$.
Prove that $d\ge p-1$.