Find the product of the sum of digits of the smallest $4$ -digit number divisible by $6$ and the sum of digits of the smallest $5$ -digit number divisible by $11$.

1. Find the smallest $4$ -digit number divisible by $6$:
* Smallest $4$ -digit number is $1000$.
* $\frac{1000}{6}=166.66⟹167\times 6=1002$.
* So, the number is $1002$. Sum of digits $=1+0+0+2=3$.

2. Find the smallest $5$ -digit number divisible by $11$:
* Smallest $5$ -digit number is $10000$.
* $\frac{10000}{11}=909.09⟹910\times 11=10010$.
* So, the number is $10010$. Sum of digits $=1+0+0+1+0=2$.

3. Product of the sums of digits:

$$\text{Product}=3\times 2=6$$