Prove that if f(x) is a polynomial with integral coefficients, and there exists a positive integer $k$ such that none of the integers f(1),f(2),...,f(k) is divisible by k, then f(x) has no integral root.

Solution:

Suppose that $f(x)$ has an integral root a. Then f(x)=(x-a)g(x) for some polynomial g(x) with integral coefficients.

Since none of the integers f(1),f(2),...,f(k) is divisible by k, none of the k consecutive integers a-1,a-2,...,a-k is divisible by k. Contradiction.