Question

If $x=5^{5^{5^{{.}^{{.}^{{.}^{.}}}}}}\left(24\text{times}5\right)$ is divided by $24$, then the remainder is

Answer 1

$x=5^{5^{5^{{.}^{{.}^{{.}^{.}}}}}}\left(24\text{times}5\right)$
As
$5^5=3125=\text{odd natural number}{5}^{5^5}=\text{odd natural number}$
Here the exponent is a odd natural number, so
$x=5^{2m+1},m\in N=5(5{)}^{2m}=5(25{)}^m=5(1+24{)}^m=5(1+{}^m{C}_1(24)+\dots +{}^m{C}_m(24{)}^m)=5(1+24k),k\in N=120k+5$

Hence, the remainder is $5$.