jmc

Any $4$ consecutive integers will have at least one multiple of $3,$ an even number not divisible by $4,$ and a multiple of $4.$ Therefore the product of any $4$ consecutive integers must be divisible by $2\cdot 3\cdot 4=\boxed{24}.$

We can check that no larger number divides every product of four consecutive integers by considering $1\cdot 2\cdot 3\cdot 4=24$.