Try writing the factorials as a product.
Now
12 = 4 x 3
1! = 1
2! = 2 x 1
3! = 3 x 2 x 1
4! = (4 x 3) x 2 x 1 <------ There is a 4 x 3 here
5! = 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here
6! = 6 x 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here
7! = 7 x 6 x 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here
Well I am sure you can see by now that 12 will divide any factorial greater than or equal to 4. That means the only fellows who can leave remainders are 1!+2!+3! = 1+2+6 = 9. So 9 is the remainder when divided by 12.
Now can you try a related question:
"Which is the largest factorial number that is not divisible by 27?"
Try it