How to solve $ 13x \equiv 1 ~ (\text{mod} ~ 17) $? 
How to solve $ 13x \equiv 1 ~ (\text{mod} ~ 17) $?

Please give me some ideas. Thank you.

Solving $13x \equiv 1 ~ (\text{mod} ~ 17)$ means there is a smallest positive integer $q$ such that we can write
$\tag 1 13q = 1 + 17k \quad \text{ for some integer } k$
But then $4k \equiv 12 ~ (\text{mod} ~ 13)$ and $k \in \{3, 16, 29, \dots\}$.
So we'll just go thru the list $[1+17\times3,1+17\times 16,\dots]$ checking for the first number divisible by $13$.
$\quad 1 + 17 \times 3 = 52 = 4 \times 13$
Well that was easy!
$\tag{ANS} 13 \times 4 \equiv 1 ~ (\text{mod} ~ 17)$
