sudai wrote:

If N is the product of all positive integers less than 31, than what is the greatest integer k for which N/18^k is an integer?

How to solve it? Pls. with explaination, thanks!

basically we need to find number of 18's available in product of 1 to 31 i.e. in 31!

or we need to find set of ( 1 two and 2 three ) in 31!

2's will be sufficiently available, so just need to count 3's

form 1,31 3's are available in 3,6,9,12,15,18,21,24,27,30

total 3's are 14 =>number of sets =14/2 (as each set require 2 3's) =7

hence there are 7 sets of (1 two and 2 three ) available in 31!

or in other words there are 7 18's available in 31! => k=7