  We prove a general divisibility theorem that implies, e.g., that, in any
group, the number of generating pairs (as well as triples, etc.) is a multiple
of the order of the commutator subgroup. Another corollary says that, in any
associative ring, the number of Pythagorean triples (as well as four-tuples,
etc.) of invertible elements is a multiple of the order of the multiplicative
group.
