Answer
$N = 30,470$
Work Step by Step
A new maximum is produced each time the mirror $L_2$ increases by a distance of $\frac{\lambda}{2}$
We can find the number of bright-dark-bright fringe shifts:
$N = \frac{\Delta L_2}{\lambda/2}$
$N = \frac{2~\Delta L_2}{\lambda}$
$N = \frac{(2)~(1.0\times 10^{-2}~m)}{656.45\times 10^{-9}~m}$
$N = 30,470$