OG-12 Number Properties Question

[Elena89]

Hi, Im unable to understand OG-12's explanation for this specific question. Can anyone please elaborate a simpler way to solve it?!

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

(A) 10
(B) 12
(C) 14
(D) 16
(E) 18

[shankar.ashwin]

The question basically asks you the number of factors of 3.
I think its 3^k and not 3k as given in the OG.

You can list it doen manually,
3
6
9 (2 3's)
12
15
18 (2 3's)
21
24
27 (3 3's)
30

So 14 factors of 3 are there in 30!

[rooster]

Hi, Im unable to understand OG-12's explanation for this specific question. Can anyone please elaborate a simpler way to solve it?!

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

(A) 10
(B) 12
(C) 14
(D) 16
(E) 18

If it's this, then E is the answer since there's enough 2s and 3s to make up 3*18=54

However, I think Shankar is right with this one, it's probably 3^k, and in that case, you need to break down all numbers into primes to see how many factors of 3 are available, in this you can list all multiples of 3, which he listed above

Then the answer is C