A unit modulo $n$ raised to $\phi(n)$ is $1$ modulo $n$.Suggest a slogan
Let \(n\ge1\) and let \(a\in\ZZ\) with \(\gcd(a,n)=1\). Then \[ a^{\phi(n)}\equiv1\pmod n, \] where \(\phi\) is the Euler totient function.
The class of \(a\) lies in the group \((\ZZ/n\ZZ)^\times\) of order \(\phi(n)\), so its order divides \(\phi(n)\) by Lagrange’s theorem.
If \(n>1\) and \(d=\gcd(a,n)>1\), then \(a^k\not\equiv1\pmod n\) for every \(k\ge1\), since \(d\) divides both \(a^k\) and \(n\) but not \(1\). For \(a=2\) and \(n=4\), \(2^{\phi(4)}=2^2=4\equiv0\pmod4\).
State Euler's theorem, and say why coprimality is needed.