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Solution: For perfect cancellation to occur, the two path lengths must differ by an odd integer multiple of $\frac{}{2}($even integer multiples instead give constructive interference$).$ Note that due to the geometry of the apparatus, if a mirror is moved by some $dx$ than the corresponding path length changes by twice that distance, $2dx.$ Thus, the mirror needs to move an odd integer multiple of $\frac{}{4}$ to give destructive interference. The smallest such value is$,$

$2$