From Ireland's Number theory book, Ch.3 ex. 6.
Let and integer $n>0$ be given. A set of integers $a_{1},a_{2}, \cdots ,a_{\phi(n)}$ is called a reduced residue system modulo $n$ if they are pairwise incongruent mod n and $(a_{i},n)=1$ for all $i$. If additionally $(a,n)=1$, prove that $aa_{1},aa_{2}, \cdots ,aa_{\phi(n)}$ is again a reduced residue system modulo n.
I can solve this problem, however, it is not clear to me how this result helps in the following (ex7)
Use ex.6 to give another proof of Euler's therorem, $a^{\phi(n)}\equiv 1 \pmod{n}$ for $(a,n)=1$