Question

If ${49}^n+16n+\lambda$ is divisible by $64$ for all $n\in N$, then the least negative integral value of $\lambda$ is

• A. $-12$
• B. $-1$
• C. $-3$
• D. $-4$

Answer 1

Answer: (B)

$-1$

For $n=1$, we have
${49}^n+16n+\lambda =49+16+\lambda =65+\lambda =64+(\lambda +1)$, which is divisible by 64 if $\lambda =-1$.
For $n=2$, we have
${49}^n+16n+\lambda ={49}^2+16\times 2+\lambda =2433+\lambda =(64\times 38)+(\lambda +1)$, which is divisible by 64 if $\lambda =-1$
Hence, $\lambda =-1$.