jmc

The prime factorization of 72 is $2^3\cdot 3^2$, which has $(3+1)(2+1)=12$ factors.

If we multiply by 2, we get $2^4\cdot 3^2$, which has $(4+1)(2+1)=15$ factors. Multiplying by any higher power of 2 gives a product with more than 16 positive factors.

Multiplying by 3 gives $2^3\cdot 3^3$, which has $(3+1)(3+1)=16$ positive factors.

Multiplying by any prime $p$ gives $2^3\cdot 3^2\cdot p$, which has $(3+1)(2+1)(1+1)=24$ factors.

Similarly, multiplying by any other positive integer besides those we've considered produces a product that has more than 16 factors. Therefore, $2^3\cdot 3^3=\boxed{216}$ is the only positive multiple of 72 with 16 positive factors.