jmc

These numbers are all in the form $n(n+1)(n+2)(n+3)(\mathrm{mod}4)$, there will be one number of each residue, so one of the numbers will be divisible by 2 and another divisible by 4, leaving the product divisible by 8. Similarly, one of the numbers will be $0\mathrm{mod}3$, so the product will be divisible by 3. The GCD must therefore be divisible by $3\cdot 8=24$. It also must be less than or equal to the smallest number in the set, $1\cdot 2\cdot 3\cdot 4=24$, so it must be $\boxed{24}$ exactly.