One main scale division of a vernier calipers is $a$ cm and nth division of the vernier scale coincide with (n-1)^th division of the main scale. The least count of the calipers (in mm) is 

According to the question.
One division of main scale reading = a cm
$n^{th}$ vernier scale division 
$=(n-1)th$  main scale divison 
$\therefore$  One division of vernier scale reading 
  $=\frac{(n-1)a}{n}\mathrm{..........}\left(i\right)$
We know that, Least count (LC)=  [ 1main scale division – 1 vernier scale division] cm 
$=a-\frac{(n-1)a}{n}$         [using Eq. (i)]
$=\frac{a(n-n+1)}{n}=\frac{a}{n}cm=\frac{a}{n}\times 10mm$

$\Rightarrow LC=\frac{10a}{n}mm$