Find the least positive residue of $5^{16} \bmod 17$

Question

I need some help on finding the least positive residues. Not sure what the correct approach is to take on these types of problems and the book I'm reading isn't helping me.

** UPDATE **

If 17 was not prime how would I solve this problem? For example, find the least positive residue of $5^{16} \bmod 18$.

Answer 1

Since $17$ is a prime, by the Fermat's little theorem we have: $$ 5^{16}\equiv 1\pmod{17}.$$

Answer 2

In this case here you can use that the exponent $16$ is a power of $2$: $2^4 = 16$:

$$ 5^{16} \equiv (5^8)^2 \equiv \ldots \equiv (((5^2)^2)^2)^2 \equiv ((25^2)^2)^2 \equiv ((8^2)^2)^2 \equiv (64^2)^2 \equiv (13^2)^2 \equiv 169^2 \equiv 16^2 \equiv (-1)^2 \equiv 1 \mod 17$$