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A major problem in knot theory is to decide whether the Jones polynomial detects the unknot. In this paper we study a weaker related problem, namely whether the Jones polynomial reduced modulo an integer m detects the unknot. The answer is known to be negative for m = 2r with r ≥ 1 and m = 3. Here we show that if the answer is negative for some m, then it is negative for mr with any r ≥ 1. In particular, for any r ≥ 1, we construct nontrivial knots whose Jones polynomial is trivial modulo 3r. [ABSTRACT FROM AUTHOR]