= Solution
The <Fermat-Euler theorem> gives
$$
a^{\varphi(N)}\equiv1\pmod N,
$$
so the <multiplicative order> $r=\operatorname{ord}_N(a)$ exists. For every <integer> $k$,
$$
f(k+r)=a^{k+r}\equiv a^ka^r\equiv a^k=f(k)\pmod N.
$$
Thus the modular exponential is <periodic modular exponentiation>[periodic] with period $r$. If $t>0$ were a smaller period, then $f(t)=f(0)$ would give $a^t\equiv1\pmod N$, contradicting the minimality of $r$. Hence $r$ is its least positive period.
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