What is the smallest number which when divided by $7$, $9$ and $11$ leaves a remainder of $4$ in each case?

The required smallest number is of the form:
$$\text{Required Number}=\text{LCM}(7,9,11)+\text{Remainder}$$

Since $7$, $9$, and $11$ are co-prime numbers (sharing no common factors except $1$), their Lowest Common Multiple ($\text{LCM}$) is the product of the numbers:

$$\text{LCM}(7,9,11)=7\times 9\times 11=63\times 11=693$$

Given that the remainder is $4$:

$$\text{Required Number}=693+4=697$$