Question

The remainder when 32 to the power 32*32 is divided by 7??PLEASE EXPLAIN IN A SIMPLE WAY??

Answer 1

$(32)^{32*32}=(35-3)^{1024}$

using binomial theorem

$(35-3)^{1024}=C_{1024}^{1024}35^{1024}+C_{1}^{1024}35^{1023}3^{1}+...+C_{0}^{1024}3^{1024}$

all the terms contain power of 35 which is multiple of 7 as 7x5=35

the last term 

$C_{0}^{1024}3^{1024}= \frac{1024!}{1024!}3^{1024}=3^{1024}$

so 

$(32)^{32*32}=(35-3)^{1024}=35k+3^{1024}=7(5k)+3^{1024}$

where k is the integer that includes the other things in expansion
so the remainder is 
$3^{1024}$