The following set of definitions comes from Discrete Mathematics with Applications by Epp S. (second edition) p. 115
An integer n is prime if, and only if, n > 1 and for all positive integers r and s, if n = r * s, then r = 1 or s = 1.
An integer n is composite if, and only if, n = r * s for some positive integers r and s with r <> 1 and s <> 1.