φ is a multiplicative function: if m and n are coprime then φ(mn) = φ(m) φ(n). (Sketch of proof: let A, B, C be the sets of residue classes modulo-and-coprime-to m, n, mn respectively; then there's a bijection between AxB and C, via the [Chinese Remainder Theorem]?.)
If n = p1k1 ... prkr where the pj are distinct primes, then φ(n) = (p1-1) p1k1-1 ... (pr-1) prkr-1. (Sketch of proof: the case r=1 is easy, and the general result follows by multiplicativity.)