The largest number that always divides the product of any four consecutive positive integers is:

Among four consecutive integers there are two even numbers, one of which is a multiple of 4, giving a factor of 8; and at least one is a multiple of 3. So $8\times 3=24$ always divides the product. Nothing bigger can be guaranteed, because the very first such product is $1\times 2\times 3\times 4=24$ itself, which is not divisible by 48 or 120.