1. Given data

• One main scale division (MSD) $= a\ \text{cm}$
• The $n^{\text{th}}$ division of the vernier scale coincides with the $(n-1)^{\text{th}}$ division of the main scale.

So, $n\,\text{VSD} = (n-1)\,\text{MSD}$ where VSD = one vernier scale division.

2. Find one vernier scale division

Using $n\,\text{VSD} = (n-1)\,\text{MSD}$ we get $\text{VSD} = \frac{n-1}{n}\,\text{MSD}$

Since $\text{MSD} = a\ \text{cm}$ therefore, $\text{VSD} = \frac{n-1}{n}a\ \text{cm}$

3. Least count formula

For a vernier calipers, $\text{Least Count} = 1\,\text{MSD} - 1\,\text{VSD}$

So, $\text{LC} = a - \frac{n-1}{n}a$

$\text{LC} = a\left(1 - \frac{n-1}{n}\right)$

$\text{LC} = a\left(\frac{1}{n}\right) = \frac{a}{n}\ \text{cm}$

4. Convert into mm

Since $1\ \text{cm} = 10\ \text{mm}$

therefore, $\text{LC} = \frac{a}{n}\times 10 = \frac{10a}{n}\ \text{mm}$

5. Match with options

$\boxed{\frac{10a}{n}}$ This corresponds to Option A.