The next theorem gives a characterization of primes.
Theorem
p | bc implies p | b
or p | c.
Let p > 1. Then p is a prime if and only if, for all integers b, c:
This theorem has the following corollary.
Corollary
If p is a prime and b1, ..., bs are integers such that p | b1 b2 ··· bs, then there is an index i
{1, ..., s} such
that p | bi.
If p is a prime and b1, ..., bs are integers such that p | b1 b2 ··· bs, then there is an index i
{1, ..., s} such
that p | bi.
We leave the proof of the corollary to the reader.