What is the smallest positive integer that is divisible by both $12$ and $16$?

To find the smallest positive integer divisible by both $12$ and $16$, we need to find their Least Common Multiple (LCM).
Let's use prime factorization:

$$12=2^2\times 3^1$$

$$16=2^4$$

Taking the highest power of each prime factor involved:

* Highest power of $2$ is $2^4$
* Highest power of $3$ is $3^1$

$$\text{LCM}=2^4\times 3^1=16\times 3=48$$

Therefore, the smallest positive integer divisible by both numbers is $48$.