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

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 $\begin{array}{l}n=2\end{array}$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, $\lambda =-1$.