Factor

[jnbimmer]

If p is the prodcut 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

[Testluv]

If p is the prodcut 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

So, the question is asking for the greatest number of 3s in p.

The easiest way to do it is to write out the multiples of 3:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30...that's 10

Because 9 is 3^2, as multiples of 9, each of 9 and 18 have one extra 3. And 27 is 3^3, so there are two extra 3s there. That's a total of 14 3s. Choice C.

[jnbimmer]

Could you explain what is the process of rephrasing the question into:

the question is asking for the greatest number of 3s in p.

Thank you!

[Pedros]

you know that P is the product of ( 1 to 30 ) inclusive, for 3^k to be a factor of P it has to include no more than the 3s included in P ; means if P includes 14 threes in its prime factoriation , 3^15 will not be a factor of P. The question is asking how many 3s are there in the prime factorization of P as explained above.

[jnbimmer]

Thanks.

Is there a shortcut to calculate the product of 1 to 30?

for example, 1x2x3x4x5...

[maihuna]

There is a formula to find such factors given a factorial:

1. Find all prime factors
2. For each factor try the number and its power till it is greater than 1.

e.g. find all multiple of 3 in !100.

Find: 100/3 100/3^2 100/3^3 100/3^4 stop now as 3^5>100

add : 33 11 3 1

Total : 48

So 3^48 is accomodable in !100

Here all multiples fro m2->30 means !30

Try : 30/3 30/9 30/27 == 10+3+1 = 14

So 3^14 is accomodable in 2-30 multiples.

[Testluv]

To be clear, the formula Maihuna used is for computing the greatest number of factors of a certain kind that can go into a particular factorial; and, if you know it, it will likely speed up solving the problem. However, I don't think it is worth it to memorize the formula; it won't have broad application on the GMAT, and in those situations where you can use it, you will almost certainly be able to use an alternative or basic technique, such as I did in this question.

[Testluv]

Thanks.

Is there a shortcut to calculate the product of 1 to 30?

for example, 1x2x3x4x5...

There is absolutely no need to calculate the product of 1 to 30!