The main scale of a vernier calliper has n divisions/cm. n...

As it is given that $\mathrm{n}$ divisions of vernier scale coincide with $(n-1)$ divisions of main scale i.e. $$\begin{aligned}& \mathrm{n}(\mathrm{VSD})=(\mathrm{n}-\mathrm{l}) \text{ MSD }\\ & \Rightarrow \quad \text{ IVSD }=\frac{(\mathrm{n}-\mathrm{l})}{\mathrm{n}} \mathrm{MSD}\\ & \end{aligned}$$The least count is the difference between one main scale division (MSD) and one vernier scale division (VSD). $$\begin{aligned}& \therefore \text{ Least Count }(\mathrm{LC})=1 \mathrm{MSD}-1 \mathrm{lVSD}\\ & =1 \mathrm{MSD}-\frac{(\mathrm{n}-\mathrm{l})}{\mathrm{n}} \mathrm{MSD}\\ & =\left(1-\frac{(\mathrm{n}-\mathrm{l})}{\mathrm{n}}\right) \mathrm{MSD}=\frac{1}{\mathrm{n}} \text{ MSD }\\ & \text{ Here, }1 \mathrm{MSD}=\frac{1}{\mathrm{n}} \mathrm{cm}\\ & \Rightarrow \quad \mathrm{LC}=\frac{\mathrm{l}}{\mathrm{n}} \times \frac{\mathrm{l}}{\mathrm{n}} \mathrm{cm}=\frac{\mathrm{l}}{\mathrm{n}^2} \mathrm{~cm}\\ & \end{aligned}$$