N divisions on the main scale of a vernier calliper coincide with...

No of divisions on main scale $=N$ No of divisions on vernier scale $=N+1$ size of main scale division $=a$ Let size of vernier scale division be $b$ then we have a N=b(N+1) \Rightarrow $\mathrm{b}=\frac{a N}{N+1}$Least count is \mathrm{a}-$\mathrm{b}=\mathrm{a}$-\frac{a N}{N+1}=a\left[\frac{N+1-N}{N+1}\right]=\frac{a}{N+1}