Question 137
Two monic polynomials (that is, with leading coefficient equal to 1) with integer coefficients $p(x)$ and $q(x)$ are such that their greatest common divisor is $(x-1)(x-2)$, their least common multiple is $(x-1)^2(x-2)^3(x-3)(x+1)$, and the degree of $p(x)$ is less than or equal to the degree of $q(x)$. In how many ways can $p(x)$ be chosen?
(A) 4
(B) 5
(C) 8
(D) 10
(E) 12
(A) 4
(B) 5
(C) 8
(D) 10
(E) 12