OG 13th ed page 163 #77 divisibility

Hi Forum,

The OG 13th ed page 163 #77 says

If n = 20! + 17, then n is divisible by which of the
following?
I. 15
II. 17
III. 19
(A) None
(B) I only
(C) II only
(D) I and II
(E) II and III


I get that 20! is divisible by 17 so adding 17 to it is still divisible by 17. I dont get the explanation they have given for 19 and 15. Can someone explian? They say :

If n were divisible by 15,
then n 20! would be divisible by 15. But,
n 20! = 17 and 17 is not divisible by 15.
Therefore, the correct answer does not include I.
If n were divisible by 19, then n 20! would be
divisible by 19. But, n 20! = 17 and 17 is not
divisible by 19. Therefore, the correct answer does
not include III.



So I decided to make my own problem with smaller numbers So I can see ... here is how I modified it

If n = 4! + 4, then n is divisible by which of the
following? (clearly in this simplified example D is the answer)
I. 4
II. 7
III. 11
(A) None
(B) I only
(C) II only
(D) I and II
(E) II and III

So basically n = 24+4 , n = 28.... n is divisible by 4 and 7 (by looking at it clearly). Now trying to back solve this by the OG's explanation

Since 4! and 4 are both divisible by 4, answer will include I. GREAT
now If n were divisible by 7 then n-4! would be divisible by 7 but n-4! = 4 . (this just like the above makes no sense with to me ... clearly 28-24 = 4 , 4 is not divisible by 7, but we know n which is 28 IS divisible by 7... is their explanation wrong ? or am I just stupid?)