Computing $16^{74} \bmod 65$ I have the following problem. I need to calculate $16^{74} \bmod 65$. Is there a way to do this without a calculator?
Thank you for your help in advance!
EDIT (M.S.): Notice that this question is linked as an example in one of the answers to the FAQ post: How do I compute $a^b\,\bmod c$ by hand?

Repeatedly raising to as small powers as possible and immediately reducing modulo $65$ we have
$$(16)^{74} = (16^2)^{37} = (256)^{37} = (260 - 4)^{37} \equiv (-4)^{37} = - 4^{37} = -4 \cdot 4^{36} = -4 \cdot (4^4)^9 = \\
-4 \cdot (256)^9 = -4 \cdot (260 - 4)^9 \equiv -4 \cdot (-4)^9 = 4^{10} = 4 \cdot (4^3)^3 = 4 \cdot (64)^3 = 4 \cdot (65 - 1)^3 \equiv \\
4 \cdot (-1)^3 = -4 \equiv 61 .$$
