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?
Pick:
A) 3 B) 6 C) 7 D) 14 E) 26
Basically the question is asking us what is the highest power of 18 that can give us a number that is a factor of 31!   So remember 18 after prime factorization is (3^2)(2). Now there are going to be less factors of 3 in 31!, than factors of 2.   Therefore, let's find how many factors of 3 in 31! We can use this quick method   31/3^1 = 10   31/3^2= 3   31/3^3=1   Sum = 14   Just ignore the remainders.   So we have that 3^14 must be the least. Now don't forget that 18 is 3^2k so k must be ONLY 7, because 2k will give us the 14.   Hence answer is (7)
<result>C</result>
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?