List all the numbers which have an inverse MOD 20

Question

Please help! I’m not sure where to start. I really need someone to thoroughly explain how to do this.

Answer 1

Let $a\in\{1,3,7,9,11,13,17,19\}$. For each of these, there exist $x,y$ such that $xa+20y=1$, by Bezout. That says that $xa\cong1\bmod{20}$. In other words, $x\cong a^{-1}\bmod{20}$.

As a check, $\varphi(20)=8$.

Answer 2

hint

$$20=2^2.5$$

$$\phi(20)=20(1-\frac 12)(1-\frac 15)=8$$

So, there are $ 8 $ inversible elements in $ \Bbb Z/20\Bbb Z$.