how to calculate ((13)^15)^17 mod 17 using fermat's little theorem?

Question

How to calculate

((13)^15)^17 (mod 17)

using fermat's little theorem?

Answer 1

As $a^{17}\equiv a\pmod{17}$ for all $a$, we only need to find $x:=13^{15}\pmod {17}$. Note that $13^2x=13^{17}\equiv 13\pmod{17}$ and as $13\equiv -4$, we have $13^2\equiv (-4)^2\equiv 16\equiv -1$, so $${(13^{15})}^{17}\equiv 13^{15}\equiv -13\equiv 4\pmod {17}. $$