# Math Help - Fermat's little theorem

1. ## Fermat's little theorem

Using Fermat's little theorem, how do i find the solutions of 7x=12(mod 17)??

i have NO clue!!!

2. Since 17 is a prime, and 17 does not divide 7, by Fermat's little theorem, we see that

$7^{17-1} = 7^{16} \equiv 1 \bmod{17}$

Then

$7^{16} = 7 \cdot 7^{15} \equiv 1 \bmod{17} \Rightarrow 7 \cdot 7^{15}\cdot 12 \equiv 12 \bmod{17}$.

So

$x = 7^{15} \cdot 12 \text{, or } x=7^{15} \cdot 12 \equiv 9 \bmod{17}$.

Hope that helps!