Solve $11^a \equiv 1 \mod 19$

Question

Find all of the possible solutions for $11^a \equiv 1 \mod 19$ . I tried to add multiples of $19$ to $1$ but it didn't work . Also Fermat's little theorem wasn't helpful .

Answer 1

By Fermat's little theorem, $11^{18} \equiv 1 \mod 19$, and so the answer must be all multiples of $d$ where $d$ is some divisor of $18$. The divisors of $18$ are $1, 2, 3, 6, 9$. Try $2$, then $3$ ...

Answer 2

By FLT we know that a solution is $a=18$ then we need also to try with $a=2,3,6,9$ and by inspection we find that $a=3$ is a solution, we say that $ord_{19}(11)=3$, and thus all $a=3k$ are solutions.