How to decide if a factorial is a multiple of certain number?

Question

How to decide if a factorial is a multiple of certain number?

For example, if I have to decide whether $123!$ is a multiple of $4$ or not what should be the procedure?

Answer 1

If the number is smaller than the factorial it certainly is a multiple. In you example, $123! = 1 \times 2 \times 3 \times 4 \times ... \times 123$. It contains 4 thus, 4 is a multiple. $123!$ can be written as $$123! = 4 \times (1 \times 2 \times 3 \times 5 \times 6 \times ... \times 123)$$

Answer 2

Every number less than $123$ and $123$ itself is a multiple of $123!$. Don't forget how $n!$ is defined, $n\times(n-1)\times(n-2)\times\dots\times1$.

Answer 3

Well in the case of $4$ being a multiple of $123!$, you just have to go back to the definition of factorial. $n! = 1\cdot2\cdot3\cdot\cdot \cdot n$ and $4$ is included there. Now for bigger numbers, you could use the fundamental theorem of arithmetic and write it as a product of it's prime factors. Then you could break the factorial into it's prime factors too which isn't difficult because if $p$ is a prime in $n!$, then $p \leq n$. For example, if you want to see if $6!$ is a multiple of $8$ then $$6!=1*2*3*4*5*6=2^43^25$$ and $$8=2^3$$ and $$2^3\mid 2^43^25$$