Question

$n\text{'}$is selected from the set $\left\{1,2,3,...,100\right\}$ and the number $2^n+3^n+5^n$ is formed. Total number of ways of selecting $\text{'}n\text{'}$ 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

$49$

Explanation for the correct option :

Finding the total number of ways of selecting $\text{'}n\text{'}$ so that the formed number is divisible by $4$, is equal to,

On dividing $2^n$ by $4$ we get $2$ as reminder when $n=1\text{and}0$of rest value of $n$

On dividing $3^n$ by $4$ we get $3$ as reminder when $n$ is odd and $1$ as reminder when $n$ is even.
On dividing $5^n$ by $4$ we get $1$ as reminder for all value of $n$.

$2^n+3^n+5^n$is divisible by $4$ only when

reminder on dividing $2^n$ by $4+$reminder on dividing $3^n$ by $4+$ reminder on dividing $5^n$ by $4$= a multiple of $4$

This is possible for all odd $n$ except $1$ because for all odd $n$ sum of reminders $=0+3+1=4$

For$n=1$ we have sum of reminder$=2+3+1=6$

So our answer will be all odd number except $1$ which is equal to $49$

Hence ,the correct answer is option (B).