Abstract: Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that the fundamental groups of nontrivial knots must have finite, non-abelian quotients. We give an explicit upper bound on the size of the smallest non-abelian quotient of the fundamental group of the complement of a nontrivial knot in S^3 with a diagram with n crossings. As motivation for such a result one should note that given a knot diagram and the corresponding Wirtinger presentation of the fundamental group of the complement, such an epimorphism onto a finite non-abelian group establishes the nontriviality of the knot.