Proofs
1. Euler's theorem can be proven using concepts from the theory of groups: The residue classes (mod n) that are coprime to n form a group under multiplication (see the article Multiplicative group of integers modulo n for details.) Lagrange's theorem states that the order of any subgroup of a finite group divides the order of the entire group, in this case φ(n). 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). But then,
2. There is also a direct proof: Let R = {x1, x2, ..., xφ(n)} be a reduced residue system (mod n) and let a be any integer coprime to n. The proof hinges on the fundamental fact that multiplication by a permutes the xi: in other words if axj ≡ axk (mod n) then j = k. (This law of cancellation is proved in the article Multiplicative group of integers modulo n.) That is, the sets R and aR = {ax1, ax2, ..., axφ(n)}, considered as sets of congruence classes (mod n), are identical (as sets - they may be listed in different orders), so the product of all the numbers in R is congruent (mod n) to the product of all the numbers in aR:
- and using the cancellation law to cancel the xis gives Euler's theorem:
Read more about this topic: Euler's Theorem
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