Question

$n$ 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

• A. $50$
• B. $49$
• C. $48$
• D. None of these

Answer 1

The correct option is B $49$

If $n$ is odd,
$3^n=4{\lambda }_1-1,5^n=4{\lambda }_2+1$
$\Rightarrow 2^n+3^n+5^n$ is divisible by $4$ if $n\ge 2$
Thus, $n=3,5,7,9,...,99$, i.e., $n$ can take $49$ 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 ${}^{\prime }{n}^{\prime }$ is equal to $49.$