Then there exists a unique natural number $b$ less than $p$ such that $ab \equiv 1 \pmod{p}$. Full question: Let $p$ be a prime and let $a$ be an integer such that $1 \leq a < p$. Then there exists a unique natural number $b$ less than $p$ such that $ab \equiv 1 \pmod{p}$.
Looking for the proof. Is Fermat's little theorem necessary?

Hint: When does a linear congruence equation $ax\equiv b($mod $m)$ have a solution?
EDIT: If you know the rule regarding division in modular arithmetic, you can find uniqueness.
