Question

$n$ is selected from the set $\{1,2,3,....100\}$ 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

Answer: (B)

$49$

On dividing $2^n$ by 4 we get 2 as reminder when $n=1$ 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 $49$