Question

Find remainder if (3232)32 is divided by 7

Answer 1

Notice that $3232 = 461*7 + 5$
Binomial theorem guarantees
$3232^{32} = (461*7 + 5)^{32} = 5^{32} + 7M$
Where $M$ is some integer. Since $7M$ is divisible by $7$ we just need to look at $5^{32}$

$5^{32} =  5^{3*10 + 2 } = 25*125^{10}= 25(18*7 - 1 )^{10}$
Using binomial theorem as before we can represent $(18*7 - 1 )^{10}$ as $(-1)^{10} + 7P$ for some integer $P$. So we just need to look at $25$ and $25 = 7*3 + 4$.