  We show that there exist infinitely many examples of pairs of knots, K_1 and
K_2, that have no epimorphism $\pi_1(S^3\setminus K_1) \to \pi_1(S^3\setminus
K_2)$ preserving peripheral structure although their A-polynomials have the
factorization $A_{K_2}(L,M) \mid A_{K_1}(L,M)$. Our construction accounts for
most of the known factorizations of this form for knots with 10 or fewer
crossings. In particular, we conclude that while an epimorphism will lead to a
factorization of A-polynomials, the converse generally fails.
