Remainder without calculator

We can write $(32^{32})^{32}$ it as, $((4 \times 8)^{32})^{32}$ or $(4^{32})^{32} \times (8^{32})^{32}$

$(4^{32})^{32} \times (7+1)^{1024}$

By Binomial Theorem,

$(4^{32})^{32} \times (7^{1024} + 1 \times 7^{1023} ... + 1^{1024})$

thus the on dividing the remainder comes out to be,

$(4^{1024})$ or $(2^{2048})$

We can observe that,

$2^{1}$ remainder by $7$ = $2$

$2^{2}$ remainder by $7$ = $4$

$2^{3}$ remainder by $7$ = $1$

$2^{4}$ remainder by $7$ = $2$

$2^{5}$ remainder by $7$ = $4$

$2^{6}$ remainder by $7$ = $1$

$2^{7}$ remainder by $7$ = $2$

$2^{8}$ remainder by $7$ = $4$

$2^{9}$ remainder by $7$ = $1$

By which we can say that $2^{2048} = 4$