Question

Remainder when $64^{32^{32}}$ is divided by 9 is equal to ______.

Answer 1

Let $32^{32} = t$.

Then,

$64^{32^{32}} = 64^t = 8^{2t} = (9 - 1)^{2t}$

Expanding using the binomial theorem, we get:
$(9 - 1)^{2t} = 9k + 1,$
for some integer $k$.

Thus, the remainder when $64^{32^{32}}$ is divided by 9 is 1.

The Correct Answer is: 1