...If integers a and b are relatively prime, then they individually divide n if and only if their product ab divides n.
This fact is well illustrated by the use Turing is said to have made of it. The sprocket wheel of his bicycle had a faulty tooth and the chain a faulty link, and unless he was pedalling very fast when the faulty parts meshed, the chain would fall off. So he counted the number, say a, of teeth on the wheel and the number, say b, of links on the chain and found that a and b were relatively prime. Between successive meetings of the bad tooth and link the sprocket wheel would in consequence go through b cycles, as the chain went through a cycles. Turing is said to have pedalled along counting, on every bth cycle of the sprocket wheel giving the burst of speed necessary to carry him past the danger point. https://dl.dropboxusercontent.com/u/43807687/math/Billingsley%20Primes%201973.pdf