A vernier calliper has a least count of C cm . It is given that N...

The least count (LC) of a vernier calliper is the difference between one main scale division (MSD) and one vernier scale division (VSD).

Given that $N$ VSD = $(N-1)$ MSD, we can write:
$1 \text{ VSD} = \dfrac{N-1}{N} \text{ MSD}$

Therefore, the least count is:
$\text{LC} = 1 \text{ MSD} - 1 \text{ VSD}$
$\text{LC} = 1 \text{ MSD} - \dfrac{N-1}{N} \text{ MSD} = \dfrac{1}{N} \text{ MSD}$

We are given that the least count is $C \text{ cm}$, so:
$\dfrac{1}{N} \text{ MSD} = C \text{ cm}$
$1 \text{ MSD} = N C \text{ cm}$

The number of divisions per cm on the main scale is the reciprocal of the length of one main scale division:
$\text{Number of divisions per cm} = \dfrac{1}{1 \text{ MSD}} = \dfrac{1}{NC}$

Answer: $\dfrac{1}{NC}$