Franklin's Notes

Euler's Theorem

Euler's Theorem states that, in the modular arithmetic ring $\mathbb Z_m$ for any $m\in\mathbb N$, $a^{\varphi(m)}=1$ for all $a\in\mathbb U_m$, the group of units of $\mathbb Z_m$. That is, for all $a\in\mathbb Z$ with $\mathrm{gcd}(a,m)=1$, we have The proof of this theorem is short and only requires a few key observations:



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