Question

If ${10}^n+3\times 4^{n+2}+\lambda$ is divisible by 9 for all $nepsilonN$, 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

(a) Let P(n) : ${10}^n+3\times 4^{n+2}+\lambda$ is divisible by 9, for all $nϵN$

For n = 1, the given statement is also true ${10}^1+{3.4}^{1+2}+\lambda$ is divisible by 9

$\because =10+3.64+\lambda =10+192+\lambda$

$=202\lambda$

If (202 + $\lambda$) is divisible by 9, then the least value of k must be 5

$\because 202+5=207$ is divisible by 9

$\Rightarrow \frac{207}{9}=23$

Hence, the least value of k is 5.