Unit's digit of the number ${36}^{1001}\times$ $7^{1002}\times$${13}^{1003}$ is

The correct option is D $8$
Unit's digit of ${36}^{1001}$ will be $6$

$7^{1002}=(7^2{)}^{501}=({49}^4{)}^{125}\times 49$

Unit's digit of $\left({49}^4\right)$ is $1$. So, unit's digit of $({49}^4{)}^{125}$ will be $1$ and thus, unit's digit of $7^{1002}$ will be $9$.

${13}^{1003}=({13}^2{)}^{501}\times 13=({169}^4{)}^{125}\times 169\times 13$

Unit's digit of ${169}^4$ is $1$. So, unit's digit of ${13}^{1003}$ will be $7$.

Hence, unit's digit of ${36}^{1001}\times 7^{1002}\times {13}^{1003}$ will be $8$.