largest factor

multiplying the known factors of this random number we get $3\times2\times2=12$

(it is very easy to see that the product of $n$ consecutive positive integers is devisible by $n!$ )

The product of any three consecutive integers is a multiple of $3$ . Since two of the numbers are even, they have prime factorizations that include $2^n$ and $2^m$ , where $m$ and $n$ are not necessarily distinct positive integers. Since the final number is odd and could be anything, none of its prime factors are known.

As such, we see that $2 \times 2 \times 3 = 12$ is the largest integer of which the random number is a definite multiple.