Lowest composite numbers (non-primes) that are NOT factors

If z=6!, what is the sum of the two lowest composite numbers (non-primes) that are NOT factors of z ?

The suggested solution is to to find integer multiples of 7, in this case 7X2 + 7X3 =35.

But why ? What theory is this ? Why select the 2nd and 3rd multiples ?

Is this generalizable ? What if I had an arbitrary number like, 36 = 3 X 12 ?

based on the understanding that 6! doesn’t have a 7 in it. So, if they didn’t mention ‘composite/non-prime’ then it’d be 7 + 14. But as per the question, the answer is the summation of two other multiples of 7. (14 + 21)
in 3*12 it’s already a multiple of 6! as it contains 2,3 and 6.

I see. Let me rephrase my question for my hypothetical because I realize it was unclear.

What if I had some arbitrary number, like 823. Is there a general way to apply this trick without brute forcing ?

i think not. i didn’t even bruteforce, it was pretty basic ig