(a) Let $p \in \mathbb{N} $ be and odd prime. Show that for $a, b \in \mathbb{N}, p ot 1 b$, the congruence $x^{a} \equiv b$ mod $p$ has solutions if and only if $b^{\frac{p-1}{h c f(a, p-1)}} \equiv 1 bmod p$. (b) Show that the congruence $x^{8} \equiv 16 \bmod p$ has solutions for all primes $p \in \mathbb{N} $. SP.SD. 1951 (a) Let $p \in \mathbb{N} $ be and odd prime. Show that for $a, b \in \mathbb{N}, p ot 1 b$, the congruence $x^{a} \equiv b$ mod $p$ has solutions if and only if $b^{\frac{p-1}{h c f(a, p-1)}} \equiv 1 bmod p$. (b) Show that the congruence $x^{8} \equiv 16 \bmod p$ has solutions for all primes $p \in \mathbb{N} $. SP.SD. 1951