It is given that $\left(N+1\right)$ Divisions on vernier scale$\left(V.S.D.\right)$=$N$ Divisions on main scale $\left(M.S.D.\right)$
$\Rightarrow 1 V.S.D.=\frac{N}{N+1}M.S.D.$

Least count of vernier callipers $\left(L.C.\right)=1 M.S.D.-1 V.S.D.$

$\Rightarrow L.C.=a-a\left(\frac{N}{N+1}\right)=\frac{a}{N+1} \mathrm{unit}$