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Say $\phi(n)=k$ is the number of integers less than n relatively prime to n. Then prove that any integer a relatively prime to n $a^{\phi(n)}=1 \quad mod\quad n$.

My proof

U(n) be the group of integers coprime to n less than n. Then a belongs to this group and order of group is k. Hence $a^k=1$.

This proof is correct right.

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If a (finite) group has order $k$ then of course every element $x$ of the group satisfies $x^k=1$; this is pure group theory.

However to apply this theorem one has to first show that U(n) is a group in the first place. SO one has to do some number-theoretic proof, there is no way to avoid it.

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