Problem Solving

What is the greatest value of x such that 3^x is a factor of 18!?

A. 9
B. 8
C. 6
D. 3
E. 2

The OA is B.

I need help to solve this PS question. Please, can any expert explain it for me? Thanks.

LUANDATO wrote:What is the greatest value of x such that 3^x is a factor of 18!?

A. 9
B. 8
C. 6
D. 3
E. 2

The prompt implies the following:
18!/(3^x) = integer.
To determine the greatest possible value of x, we need to count how many 3's can divide into 18!.
Put another way:
We need to count how many 3's are contained within 18!.

The following multiples of 3 are contained with 18!:
3, 5, 9, 12, 15, 18.
If we factor out all of the 3's contained within the values above, we get:
3
6=3*2
9=3*3
12=3*4
15=3*5
18=3*3*2.

The blue values indicate that there are eight 3's contained within 18!.
Implication:
Up to eight 3's can divide into 18!.
Thus, the greatest possible value of x = 8.

The correct answer is B.