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

The smallest positive integer divisible by two numbers is their Least Common Multiple (LCM).
Finding prime factorizations:

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

$$16=2^4$$

To calculate the LCM, take the highest power of all prime numbers involved:

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

Therefore, the smallest positive integer divisible by both $12$ and $16$ is $48$.