Need help with OG13 quant q116

[cml907]

The question reads "If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer for k for which 3^k is a factor of p?"
a) 10
b) 12
c) 14
d) 16
e) 18

I am kind of comfortable with the explanation of finding all the multiples of 3 between 1 and 30 but I don't understand how you can sum the number of factors of 3 (see problem explanation on pg 225) to get k. How do you know that the number for k will be a factor of p with this solution method?

Any help is appreciated! Thanks so much!

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Hi cml907,

These types of questions are based on a math concept called "prime factorization", which basically means that any integer greater than 1 is either prime OR the product of a bunch of primes.

Here's a simple example:

24 = (2)(2)(2)(3)

Now, when it comes to this question, we're asked to multiply all the integers from 1 to 30, inclusive and find the greatest integer K for which 3^K is a factor of this really big number.

Here's a simple example with a smaller product:
1 to 6, inclusive...
(1)(2)(3)(4)(5)(6)

Then numbers 1, 2, 4 and 5 do NOT have any 3's in them, so we can essentially ignore them:
3 = one 3
6 = (2)(3) = one 3
Total = two 3's

So 3^2 is the biggest "power of 3" that goes into the product of 1 to 6, inclusive.

Using that same idea, we need to find all of the 3's in the product of 1 to 30, inclusive. Here though, you have to be careful, since there are probably MORE 3's than immediately realize:

3 = one 3
6 = one 3
9 = (3)(3) = two 3s
12 = one 3
15 = one 3
18 = (2)(3)(3) = two 3s
21 = one 3
24 = one 3
27 = (3)(3)(3) = three 3s
30 = one 3

Total = 14 3's

Final Answer: C

GMAT assassins aren't born, they're made,
Rich