If $p$ is a prime and $a,b$ are integers not divisible by $p$ such that $a^x \equiv b \pmod p$ with $0 ≤ x < o_p(a)$, then we define $x = L_a(b)$ and say $x$ is the discrete logarithm of $b$ with respect to $a$ (of course the discrete log also depends on $p$). Suppose $o_p(a)=r$. Prove that for any integer $b$ not divisible by $p$, the discrete log $L_a(b)$ exists if and only if $b^r \equiv 1 \pmod p$.
[ $o_p(a)$ is the (multiplicative) order or $a$ modulo $p$. ]
Observe the definition of Discrete Logarithm
$ord_pa=r\iff a^r\equiv1\pmod p$
$b^r\equiv(a^x)^r\equiv (a^r)^x\equiv 1^x\equiv1\pmod p$
Conversely, if $b^r\equiv1\pmod p\implies (a^x)^r\equiv1\implies a^{rx}\equiv1$
As $a,b$ are arbitrary so will be $x$ then $a^{rx}\equiv1\pmod p$ will be true iff $r=p-1$