cWhat is the importance of Multiplicative Group in Number Theory

Question

I'm studying Number theory basics for Cryptography Course. There is a term called Multiplicative Group which confuses me litle bit

I know $|\mathbb Z_n^*| = \phi(n)$ (Euler Phi Function) and

$\mathbb Z_n^* = \{ a \in \mathbb Z_n | gcd(a,n) = 1 \}$

I think it has to do something with inverses, But I'm not sure. Can anybody explain this group structure. How is it helpful and where ..

Answer 1

One of the requirements for a group is that every element has an inverse. For $\mathbb{Z}_n$ groups under multiplication, the only elements that have multiplicative inverses are those which are relatively prime to $n$. Or in symbols, $a \in \mathbb{Z}_n^*$ if and only if $\gcd(a,n) = 1$. Euler's phi function comes into play because it counts all such $a$.