Let $a,m\in\mathbb{N^*}$, with $m>1$, such that $(a,m)=1$. Show that, if $n_1\equiv n_2\pmod{\phi(m)}$, then $a^{n_1}\equiv a^{n_2}\pmod m$

Question

Let $a,m\in\mathbb{N^*}$, with $m>1$, such that $(a,m)=1$. Show that, if $n_1\equiv n_2\pmod{\phi(m)}$, then $a^{n_1}\equiv a^{n_2}\pmod m$.

I thought about some things, but nothing conclusive, I thought of using the Euler's theorem, or check $m\mid \phi(m)$ in the case, but nothing helped me, could save me?

Answer 1

WLOG let's $n_1 \ge n_2$

From the condition we have: $n_1 = \phi(m)\cdot k + n_2$, where $k \in N$

Then we have: $a^{n_1} = a^{\phi(m)\cdot k + n_2} = a^{\phi(m)\cdot k} \cdot a^{n_2} \equiv a^{n_2} \pmod m$

All you need is to know Euler's Theorem.