Please help! I’m not sure where to start. I really need someone to thoroughly explain how to do this.
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Hint: In $\mathbb{Z}$ mod $p$, the element $[x]$ has an inverse if and only if $\gcd(x,p) = 1$. Which elements have that property mod $20$? – Brad G. Sep 30 '20 at 23:16
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2By the linked dupes $,a,$ is invertible $\bmod 20\iff \gcd(a,20)=1\iff \gcd(a,2)=1=\gcd(a,5),, $ i.,e. $,a,$ is odd and not a multiple of $5\ \ $ – Bill Dubuque Sep 30 '20 at 23:58
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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, $\varphi(20)=8$.
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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$.
hamam_Abdallah
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