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As stated previously, factorials have A LOT of factors. So how do we find values that are NOT factors of a factorial? Let's say for example we're trying to find the smallest integers that are not factors of $20!$.
- Step 1: Simply find the prime numbers greater than $20$.
$23, 29, 31, 37, 41$, etc. are all NOT factors of $20!$
- Step 2: But what if we want to find the non-prime non-factors? In this case, you simply need to find the multiples (greater than the number itself) of our previously identified primes:
$23$: $46, 69, 92...$
$29$: $58, 87, 116...$
You get the idea. NOTE that this trick really only works for factorials greater than or equal to $10!$. If it's a small factorial, you're better off finding the factors using a more brute-force, list them all out approach.