MGMAT - Factor

Anurag@Gurome wrote:

diehard_gmat wrote:If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18

p = 1*2*3*...*29*30 = 30!

The problem is basically asking us to find the number of 3's in p.
Now all the multiples of 3 (3, 6, ... , 30) has one 3 in them.
Number of multiples of 3 within 30 = 10

But 9 and 18 each has one extra 3 in them and 27 has two extra 3 in it.

Hence, total number of 3's in p = 10 + 1 + 1 + 2 = 14

The correct answer is C.

I got up to 30!, but then I didn't know what to do. 30! is such a huge number. How were you able to tell that the question wants to know the number of integers within 1 though 30 that has only one factor of three?

greatest integer k signals some highest common factor and i wish to preform prime factorization of 30! for the number of 3s
30*29*......
all in all we have 3*10 in 30 One three, 3*9 in 27 Three threes, 3*8 One three, 3*7 One three, 3*6 Two threes, 3*5 On three, 3*4 One three, 3*3 Two threes, and 3*1 One three --> total make-up 1+3+1+1+2+1+1+2+1+1=14, hence k=14

IOM C

diehard_gmat wrote:If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18