mod of a number with a really long exponent

Question

How would I go about calculating this quickly without a calculator?

$| 157^{888888888888888888889} | \mod(15) $

I don't see a way to use small fermat theorem.

Eulers thing leaves me with the big number $\mod(8)$. Not that much easier.

dividing the thing up still leaves me with an array of 1s..

Answer 1

Hint: That big number $\mod 8$ is absurdly easy to calculate.

Answer 2

"Euler's thing" $\phi(15) = 8$

A really big number mod 8

$888888888889\equiv 1\pmod 8$

And off the top we can say $157^k \equiv 7^k\pmod {15}$

Where are we?

$15^{888888888889}\equiv 7^1\pmod {15}$