How many invertible elements are there in $\mathbb {Z_ {990}}$ and $\mathbb {Z_ {1060}}$. How many invertible elements are there in $\mathbb {Z_ {990}}$ and $\mathbb {Z_ {1060}}$ . Justify your answer.
Hello, could someone explain to me how to find the invertible elements of a set as large as $\mathbb {Z_ {990}}$ and $\mathbb {Z_ {1060}}$
I thought about putting together the set and trying each one but it is very big and it can take forever

Let $\varphi(n) = \{a\in \mathbb{Z}_n : a \text{ is invertible}\} = \{a\in \mathbb{Z}_n : \gcd(a,n) = 1\}.$
If $n=p_1^{e_1} \cdots p_s^{e_s}$ is the prime factorization of $n$, then (one can prove that) $$\varphi(n) = n\prod_{k=1}^{s}\Big(1-\dfrac{1}{p_k}\Big).$$
For $n=990,$ we have that $n=2\cdot3^2\cdot5\cdot11$, so $$\varphi(990)=990\cdot\Big(1-\dfrac{1}{2}\Big)\cdot\Big(1-\dfrac{1}{3}\Big)\cdot\Big(1-\dfrac{1}{5}\Big)\cdot\Big(1-\dfrac{1}{11}\Big)=240.$$
