Question

If ${10}^n+3\cdot 4^{n+2}+\lambda$ is exactly divisible by $9$ for all $n\in N$, then the least positive integral value of $\lambda$ is

• A. $5$
• B. $3$
• C. $7$
• D. $1$

Answer 1

Answer: (A)

$5$

Let $P\left(n\right)={10}^n+3\cdot 4^{n+2}+\lambda$
Since $P\left(n\right)$ is divisible by $9$ for all $n\in N$
$\Rightarrow P\left(1\right)$ will be also divisible by 9
$\Rightarrow P\left(1\right)=10+3\left(4^3\right)+\lambda =202+\lambda =22\times 9+(\lambda +4)$
Thus least positive integral value of $\lambda$ for which $P\left(1\right)$ can be divisible by $9$ is $5$
Hence option 'A" is correct choice.