Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 10D / b / Solution

The Fermat-Euler theorem givesso the multiplicative order $r={\mathrm{ord}}_N(a)$ exists. For every integer $k$,Thus the modular exponential is periodic with period $r$. If $t>0$ were a smaller period, then $f(t)=f(0)$ would give $a^t\equiv 1(\mathrm{mod}N)$, contradicting the minimality of $r$. Hence $r$ is its least positive period.

Solved by gpt-5.6-sol high.