stonecold wrote:

What is the greatest possible value of integer n if 100! is divisible by 15^n

A)20

B)21

C)22

D)23

E)24

We need to determine the largest value of n such that 100! is divisible by 15^n. For a number to be divisible by 15, it must be divisible by both 3 and 5. Thus, we need to find the largest value of n such that 100! is divisible by 3^n x 5^n.

Since we know there are fewer 5s in 100! than 3s, we can find the number of 5s and thus be able to determine the number of 5-and-3 pairs.

To determine the number of 5s within 100!, we can use the following shortcut in which we divide 100 by 5, then divide the quotient of 100/5 by 5 and continue this process until we no longer get a nonzero quotient.

100/5 = 20

20/5 = 4

Since 4/5 does not produce a nonzero quotient, we can stop.

The final step is to add up our quotients; that sum represents the number of factors of 5 within 100!.

Thus, there are 20 + 4 = 24 factors of 5 within 100!

Since there are 24 factors of 5 within 100!, we also know that there are 24 5-and-3 pairs and, thus, the largest value of n is 24.

Answer: E