Suppose $gcd(a,n)=1$. If $a^x\equiv b\pmod n$ and $xy\equiv 1\pmod {\phi(n)}$, show that $a\equiv b^y\pmod n$.

Question

My midterm exam is coming and I have some problem in dealing with this kind of question. This is an exercise on my text book and not a homework.

Suppose $gcd(a,n)=1$.

Question(a)

If $a^x\equiv b\pmod n$ and $xy\equiv 1\pmod {\phi(n)}$, show that $a\equiv b^y\pmod n$.

Question(b)

If $gcd(x,{\phi(n)})=1$, show that $a^x\equiv b^y$ if and only if $a\equiv b\pmod n$.

Answer 1

$$a^x\equiv b\pmod n\implies b^y\equiv a^{xy}\equiv a^1\pmod n$$ as $xy\equiv1\pmod{\phi(n)}$