Given $n=p_1^{k_1},...,p_r^{k_r}$. Show that $a^{lcm(\phi{(p_1^{k_1})},\phi{(p_2^{k_2})},...,\phi({p_r^{k_r})})}\equiv 1\pmod n$ I am doing revision for my final exam. This is a sample question given by TA.
I actually have some idea, but wondering it is true or not.
I guess $lcm(\phi{(p_1^{k_1})},\phi{(p_2^{k_2})},...,\phi({p_r^{k_r})})$ here is just equal to $\phi{(n)}$, right? Then, it is just another form of Euler's Theorem.
Can anyone answer this for me?

It is not the case that $\text{lcm}(\phi(p_1^{k_1}), \ldots, \phi(p_r^{k_r}))$ is necessarily equal to $\phi(n)$:  Take $n=12$, for instance.  In order to prove your statement, try showing that the congruence holds modulo each of the prime powers $p_i^{k_i}$ using Euler's theorem for those numbers.
