Question

If ${10}^n+{3.4}^{n+2}+\lambda$ is exactly divisible by 9 for all $nϵN$, then the least positive integral value of $\lambda$ is

• A. $5$
• B. $3$
• C. $7$
• D. $1$

Answer 1

The correct option is A $5$

Let $P\left(n\right)={10}^n+3.4^{n+2}+\lambda =S_n$ is divisible by 9

$\Rightarrow {10}^n+3.4^{n+2}+\lambda$ is divisible by 9.

$P\left(1\right)={10}^1+3.4^{1+2}+\lambda$ is divisible by 9

$\Rightarrow \lambda +202$ is divisible by 9.

Therefore, the least value of $\lambda$ is 5.

(Note: $5+202=207$ is divisible by 9)