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@moth
/pedant
that condition is insufficient, as it could be met by ratios such as 13:52, 17:51, 17:34, 19:57... i.e. where the larger sprocket has the smaller prime sprocket for a factor. Coprime is the right answer, and two integers are coprime if they share no prime factors. Prime guarantees coprime only if it's the larger that is prime or the ratio is closer than 2:1.