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The order of any subgroup of a finite group divides the order of the entire group. If a is any number coprime to n then a is in one of these residue classes, and its powers a, a2, ..., ak ≡ 1 (mod n) are a subgroup. Lagrange's theorem says k must divide φ(n), i.e. there is an integer M such that kM = φ(n).
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