Problem

Source: 2011 Saudi Arabia IMO TST 3.3

Tags: algebra, polynomial, number theory, divides



Let $f \in Z[X]$, $f = X^2 + aX + b$, be a quadratic polynomial. Prove that $f$ has integer zeros if and only if for each positive integer $n$ there is an integer $u_n$ such that $n | f(u_n)$.