Which is the smallest positive integer having exactly 12 positive divisors?

If $n=p^a\times q^b$ × …, the divisor count is $(a+1)(b+1)$…, so we need that product to be 12. Splitting 12 as $3\times 2\times 2$ gives exponents 2, 1, 1 on the three smallest primes: $2^2\times 3\times 5=60$. Splitting as $4\times 3$ gives $2^3\times 3^2=72$, and as $6\times 2$ gives $2^5\times 3=96$ — both valid but larger. The odd one out is $48=2^4\times 3$, which has $(4+1)(1+1)=10$ divisors, not 12.