Summary: Define the \(\text{MOD}_m\)-degree of a boolean function \(F\) to be the smallest degree of any polynomial \(P\), over the ring of integers modulo \(m\), such that for all \(0 - 1\) assignments \(\vec x\), \(F (\vec x) = 0\) iff \(P (\vec x) = 0\). We obtain the unexpected result that the \(\text{MOD}_m\)-degree of the OR of \(N\) variables is \(O (\root r \of {N})\), where \(r\) is the number of distinct prime factors of \(m\). This is optimal in the case of representation by symmetric polynomials. The \(\text{MOD}_n\) function is 0 if the number of input ones is a multiple of \(n\) and is one otherwise. We show that the \(\text{MOD}_m\)-degree of both the \(\text{MOD}_n\) and \(\neg \text{MOD}_n\) functions is \(N^{\Omega (1)}\) exactly when there is a prime dividing \(n\) but not \(m\). The \(\text{MOD}_m\)-degree of the \(\text{MOD}_m\) function is 1; we show that the \(\text{MOD}_m\)-degree of \(\neg \text{MOD}_m\) is \(N^{ \Omega (1)}\) if \(m\) is not a power of a prime, \(O(1)\) otherwise. A corollary is that there exists an oracle relative to which the \(\text{MOD}_mP\) classes (such as \(\oplus P)\) have this structure: \(\text{MOD}_mP\) is closed under complementation and union iff \(m\) is a prime power, and \(\text{MOD}_nP\) is a subset of \(\text{MOD}_mP\) iff all primes dividing \(n\) also divide \(m\).