Is this a true conclusion? "Except 1 and 2, any numbers factorial is divisible by 3"

Question

hi all, I've observed some kind of pattern on factorial values plenty integer numbers and I observed two patterns that I don't know before.

1. except 1! and 2!, all of my tests resulted​ on some number divisible by 3. example: 5!==120, 120%3==0

2. except 1! and 2!, the sum of the digits of the factorial values are divisible by 3 too.
example: 5!==120, 1+2+0==3, 3%3==0.

Answer 1

Well, yes. A factorial of a natural number x where x >2 means that 3 will be used in the factorial of x. (3 will be multiplied by every other natural number up to x) So when you are testing if it is a divisor, it must be, because you are just dividing that 3 out that you multiplied in earlier. Hope my reasoning makes sense..🖒

Answer 2

J.G. Yes, your reasoning is Awesome!
Your answer nails it!
Haha...!

Answer 3

I see. any factorial value is the product of consecutive natural numbers​. hence, divisible by these natural numbers.! I love sololearn!

Answer 4

good conclusion @jonas

Answer 5

I want to add something. Your second point is connected with your first one. A number whose digit's sum is divisible by 3 has to be divisible by 3 too.