Moving the mirror a distance of $\lambda_{\text{red}}/2$ causes the fringe pattern to shift by a distance equal to the distance between two fringes. Therefore, if the fringe pattern shifts so that $100,000$ fringes pass across the detector for green light, the mirror must have moved a distance

$\begin{align*}d = \frac{\lambda_{\text{green}}}{2} \cdot 100,000\end{align*}$

The same argument can be applied to the red laser, therefore
$\begin{align*}d = \frac{\lambda_{\text{red}}}{2} \cdot 85,865\end{align*}$
We can combine the two equations above to give:

$\begin{align*}\frac{\lambda_{\text{green}}}{2} \cdot 10^5 = \frac{\lambda_{\text{red}}}{2} \cdot 85865 \Rightarrow \lambda_{\...$
Therefore, answer (B) is correct.

Solution to 2001 Problem 100