CMO 1977 Problem 4

Let $$p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0$$ and $$q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0$$ be two polynomials with integer coefficients. Suppose that all the coefficients of the product $p(x) \cdot q(x)$ are even but not all of them are divisible by 4. Show that one of $p(x)$ and $q(x)$ has all even coefficients and the other has at least one odd coefficient.