A Congruence for Products of Binomial Coefficients modulo a Composite

For a positive composite integer n, we investigate the residues of ( mn k ) for positive integers m and k. First, we discuss divisibility of such coefficients. Then we study congruence identities relating products of binomial coefficients modulo n. Certainly the Chinese Remainder Theorem can be used in combination with prime power results to evaluate binomial coefficients modulo a composite. However, in this study we investigate residues modulo a composite directly without appealing to the Chinese Remainder Theorem. And as a result we get closed form identities modulo a composite. One of the many consequences of the main result (Theorem 8) is the following: If p, q and r are primes and m is a positive integer, then mr ( mpqr pq ) ≡ ( mqr q )( mpr p ) (mod pqr). Several numerical examples of these results are included.