In a vernier caliper, $N$divisions of the main scale coincide with $N + m$ divisions of the vernier scale. What is the value of $m$ for which the instrument has minimum least count?

In a vernier calipers the main scale and the vernier scale are made up different materials. When the room temperature increases by $\mathrm{\Delta }T°\mathrm{C}$, it is found the reading of the instrument remains the same. Earlier it was observed that the front edge of the wooden rod placed for measurement crossed the $N^{\mathrm{th}}$ main scale division and $N+2 \mathrm{MSD}$  coincided with the $2^{\mathrm{nd}} \mathrm{VSD}$. Initially, $10 \mathrm{VSD}$  coincided with$9 \mathrm{MSD}$. If coefficient of linear expansion of the main scale is ${\alpha }_1$ and that of the vernier scale is ${\alpha }_2$ then what is the value of $\frac{{\alpha }_1}{{\alpha }_2}$ ? (Ignore the expansion of the rod on heating)