We know that there is a one-to-one correspondence between the $r$ with $0\le r\lt20$ with $gcd(r,20)=1$ and the pairs $(r_1,r_2)$ where $0\le r_1\lt4$, where $0\le r_2\lt5$ such that $gcd(r_1,4)=1$ and such that $gcd(r_2,5)=1$. Exhibit this correspondence.
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Perhaps you should read about the Chinese Remainder Theorem. You may also find it helpful to take some specific pairs (r,s) (of integers modulo 4 and 5, respectively) and "assemble" them to form a residue class modulo 20.