 Solution

Choose the representative $1\le a\le p-1$. Since $p$ is prime and does not divide $a$, their greatest common divisor is one. The extended Euclidean algorithm gives the Bezout identityfor some integers $x,y$. Reducing modulo $p$ gives $ax\equiv 1(\mathrm{mod}p)$, so the residue class of $x$ is a multiplicative inverse of $a$. The argument uses the nonzero condition to exclude the only residue with greatest common divisor $p$.