Question

Let $n$ is selected from the set $\{1,2,3,...,10\}$ and the number $2^n+3^n+5^n$ is formed. Total number of ways of selecting $n$ so that the formed number is divisible by $4$ is equal to

Answer 1

$3^n=4{\lambda }_1-1,5^n=4{\lambda }_2+1$
If $n$ is odd,
$\Rightarrow 2^n+3^n+5^n$ is divisible by 4 if $n\ge 2$
Thus, $n=3,5,7,9$, i.e., $n$ can take 4 different values.
If $n$ is even, $3^n=4{\lambda }_1-1,5^n=4{\lambda }_2+1$
$\Rightarrow 2^n+3^n+5^n$ is not divisible by 4
as $2^n+3^n+5^n$ will be in the form of $4\lambda +2.$
Thus, the total number of ways of selecting '$n$' is equal to $4$.