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

To find: the remainder of  $$64^{32^{32}}$$

We know $$9\times\ 7=63$$

$$\therefore$$ When we divide $$\frac{64}{7}$$ we get remainder 1.

So we can also write $$64^{32^{32}}$$ as $$1^{32^{32}}$$

When we divide $$1^{32^{32}}$$ by 9 we get a remainder 1.

$$\therefore$$ correct answer is 1.

ALTERNATE METHOD:

Simplify 64 mod 9

$$64=\left(9\times\ 7\right)+1$$

So, 64 ≡1 (mod 9) 

Now raise both side to the power $$32^{32}$$

$$\therefore$$ $$64^{32^{32}}≡1^{32^{32}}$$ (mod 9)

Since any power of 1 is 1 

$$\therefore$$ $$64^{32^{32}}$$ is divided by 9 the remainder is 1.