The least count of a vernier caliper is y . The length of one main...

The least count (L.C.) of a vernier caliper is given by the difference between one main scale division (MSD) and one vernier scale division (VSD).

Given that $1 \text{ MSD} = x$ and $\text{L.C.} = y$.

Since $N$ divisions of the vernier scale coincide with $M$ divisions of the main scale, we have:
$N \text{ VSD} = M \text{ MSD}$
$\Rightarrow 1 \text{ VSD} = \dfrac{M}{N} \text{ MSD} = \dfrac{M}{N} x$

Now, substitute this into the least count formula:
$\text{L.C.} = 1 \text{ MSD} - 1 \text{ VSD}$
$y = x - \dfrac{M}{N} x$

Rearranging the terms to solve for $N$:
$\dfrac{M}{N} x = x - y$
$\dfrac{M}{N} = \dfrac{x - y}{x}$
$N = \dfrac{Mx}{x - y}$

Answer: $\dfrac{Mx}{x-y}$