jmc

Recall that we can determine the number of factors of $n$ by adding $1$ to each of the exponents in the prime factorization of $n$ and multiplying the results. We work backwards to find the smallest positive integer with $5$ factors. Since 5 is prime, the only way for a positive integer to have 5 factors is for the only exponent in its prime factorization to be 4. The smallest fourth power of a prime number is $2^4=\boxed{16}$.