2. Positive Integer P has exactly 2 positive prime factors, 5 and 11. If P has a total of 8 positive factors, including 1 and P, what is the value of P?
(1) 5^3 is a factor of P
(2) 11^2 is not a factor of P
We can break down the prime factorization of P as follows:
P = 5^a * 11^b (where a and b are positive integers)
There's a handy little trick whereby you can find the number of factors of any number using its prime factorization. Simply bump each exponent up by 1 and multiply the results:
(a+1)*(b+1) = 8
There are only two ways in which this could work:
a=1, b=3
OR
a=3, b=1
Therefore, P is either 5^3 * 11 or 5 * 11^3.
Statement (1) says 5^3 is a factor of P. Therefore, P = 5^3 * 11. SUFFICIENT
Statement (2) says 11^2 is not a factor of P. That means P cannot be 5 * 11^3, and thus it must be 5^3 * 11. SUFFICIENT
If you're interested in learning more about the prime-factorization number-of-factors technique, check out this blog post I wrote:
https://blog.knewton.com/2010/05/06/gmat ... d-factors/
Hope this helps!
Problem 2-DS
This topic has expert replies
Source: Beat The GMAT — Data Sufficiency |
-
Rich@VeritasPrep
- GMAT Instructor
- Posts: 147
- Joined: Tue Aug 25, 2009 7:57 pm
- Location: New York City
- Thanked: 76 times
- Followed by:17 members
- GMAT Score:770
raz1024 wrote:2. Positive Integer P has exactly 2 positive prime factors, 5 and 11. If P has a total of 8 positive factors, including 1 and P, what is the value of P?
(1) 5^3 is a factor of P
(2) 11^2 is not a factor of P
We can break down the prime factorization of P as follows:
P = 5^a * 11^b (where a and b are positive integers)
There's a handy little trick whereby you can find the number of factors of any number using its prime factorization. Simply bump each exponent up by 1 and multiply the results:
(a+1)*(b+1) = 8
There are only two ways in which this could work:
a=1, b=3
OR
a=3, b=1
Therefore, P is either 5^3 * 11 or 5 * 11^3.
Statement (1) says 5^3 is a factor of P. Therefore, P = 5^3 * 11. SUFFICIENT
Statement (2) says 11^2 is not a factor of P. That means P cannot be 5 * 11^3, and thus it must be 5^3 * 11. SUFFICIENT
If you're interested in learning more about the prime-factorization number-of-factors technique, check out this blog post I wrote:
https://blog.knewton.com/2010/05/06/gmat ... d-factors/
Hope this helps!
Thanks a ton!
This trick is indeed handy and it helps. I will look into other problems like this so that I can apply this kind of approach.
