I recently learnt modular arithmetic for finding remainders when huge numbers are to be divided by some number. However, I am stuck at this problem:
What is the remainder when $\displaystyle32^{32^{32}}$ is divided by $7$?
I suppose the idea here is to reduce the exponent $32^{32}$ to such a number that $32$ raised to $32^{32}$ and $32$ raised to that number will give the same remainder. But how do we do that?
${\rm mod}\ 3\!:\ 32^{32}\equiv (-1)^{32}\equiv 1,\ $ so $\ \color{#c00}{ 32^{32} = 1 + 3k}$
${\rm mod}\ 7\!:\ 32^{\large 32^{32}}\!\equiv 4^{\large \color{#c00}{32^{32}}}\!\equiv 4^{\large \color{#c00}{1+3k}}\equiv 4(4^{\large 3})^{\large k}\equiv 4(1)^{\large k}\equiv 4 $
Note that $32 \equiv 4$ mod $7$ so we can equally well work with $4^{32^{32}}$.
$4^3=64\equiv 1$ mod $7$ so we need to look at the exponent $32^{32}$ modulo $3$ - i.e. adding or subtracting $3$ to the exponent doesn't change the remainder on division by $7$.
Note that $2^2=4\equiv 1$ modulo 3, and so working mod $3$ we have $$32^{32}\equiv2^{32}=(2^2)^{16}\equiv 1^{16}=1$$
So, working mod $7$ we have $$32^{32^{32}}\equiv4^{32^{32}}\equiv 4^1=4$$
First show that the exponent $32^{32} \cong 2 \pmod 6$. Then $32^2 \cong 2 \pmod 7$ (by Fermat's Little theorem).
Observe $2^3 \equiv 1$(mod 7). So $(32)^{3k} \equiv 1$ (mod 7). $32 = 3 \times 10 + 2$. So $$ {32}^ {{32}^{32}} \equiv {32}^{(3 \times 10 + 2)^{32}} \equiv 32^{2^{32}} \equiv 32^{3k+1} \equiv 32 \equiv 4 \; (mod \; 7)$$
