GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 21 Jan 2019, 22:47

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

## Events & Promotions

###### Events & Promotions in January
PrevNext
SuMoTuWeThFrSa
303112345
6789101112
13141516171819
20212223242526
272829303112
Open Detailed Calendar
• ### The winners of the GMAT game show

January 22, 2019

January 22, 2019

10:00 PM PST

11:00 PM PST

In case you didn’t notice, we recently held the 1st ever GMAT game show and it was awesome! See who won a full GMAT course, and register to the next one.
• ### GMAT Club Tests are Free & Open for Martin Luther King Jr.'s Birthday!

January 21, 2019

January 21, 2019

10:00 PM PST

11:00 PM PST

Mark your calendars - All GMAT Club Tests are free and open January 21st for celebrate Martin Luther King Jr.'s Birthday.

# For a certain positive integer N, N^3 has exactly 13 unique factors. H

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 52348
For a certain positive integer N, N^3 has exactly 13 unique factors. H  [#permalink]

### Show Tags

18 Feb 2017, 01:49
2
14
00:00

Difficulty:

65% (hard)

Question Stats:

50% (01:36) correct 50% (01:35) wrong based on 228 sessions

### HideShow timer Statistics

For a certain positive integer N, N^3 has exactly 13 unique factors. How many unique factors does N have?

A. 1
B. 2
C. 3
D. 4
E. 5

_________________
CEO
Joined: 11 Sep 2015
Posts: 3355
Re: For a certain positive integer N, N^3 has exactly 13 unique factors. H  [#permalink]

### Show Tags

18 Feb 2017, 07:19
1
Top Contributor
5
Bunuel wrote:
For a certain positive integer N, N³ has exactly 13 unique factors. How many unique factors does N have?

A. 1
B. 2
C. 3
D. 4
E. 5

IMPORTANT:
If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40

-------now onto the question------------------------------------------------

In the above rule, notice that total number of divisors is a PRODUCT [e.g., (4+1)(3+1)(1+1) =(5)(4)(2) = 40]

In this question, we're told that there are 13 divisors.
There's only ONE WAY that the number 13 can be written as a product: 1 x 13

So, a number with 13 positive divisors must have a prime factorization that looks like this: prime^12

So, if N³ is equivalent to prime^12, then N must equal prime^4, since (prime^4)^3 = prime^12

For example, if N = 2^4, then N = 16
Notice that, if N = 2^4, then N³ = (2^4)^3 = 2^12
And, from the above rule, if N³ = 2^12, then the number of positive divisors of N³ = (12+1) = 13

How many unique factors does N have?
If N = 16 (or any other prime^4), then the factors of 16 = {1, 2, 4, 8, 16}
There are 5 factors.

RELATED VIDEO FROM OUR COURSE

_________________

Test confidently with gmatprepnow.com

##### General Discussion
Senior Manager
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
Re: For a certain positive integer N, N^3 has exactly 13 unique factors. H  [#permalink]

### Show Tags

18 Feb 2017, 02:10
3
Bunuel wrote:
For a certain positive integer N, N^3 has exactly 13 unique factors. How many unique factors does N have?

A. 1
B. 2
C. 3
D. 4
E. 5

13 - prime, hence we have only one possible option - $$p^{12}$$

$$p^{12} = (p^4)^3$$

$$N = p^4$$, where $$p$$ is prime.

$$N$$ has 5 distinct factors.

Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1322
Location: Malaysia
For a certain positive integer N, N^3 has exactly 13 unique factors. H  [#permalink]

### Show Tags

20 Mar 2017, 00:30
1
Bunuel wrote:
For a certain positive integer N, $$N^3$$ has exactly 13 unique factors. How many unique factors does N have?

A. 1
B. 2
C. 3
D. 4
E. 5

OFFICIAL SOLUTION

The shortest way to solve this problem is by applying the Unique Factors Trick. That trick is a tool that enables the direct calculation of the total number of factors of any number based on that number’s prime factor list. Specifically, to count the total factors of any number, prime factor the number, discard the bases, add one to the exponents, and then multiply the values obtained. The result is the total factor count.

In the case at hand, we’re told that $$N^3$$ has exactly 13 factors. When we consider the Unique Factors Trick, we can see that 13 must be the result of multiplying a set of positive integers. However, 13 is prime; its only factors are 1 and 13. So our original exponents, before adding one to each, can only have been zeroes and a single 12. In other words, $$N^3$$ must be the 12th power of a prime number:

$$N^3=p^{12}$$

From there, we can take the cube root and find that N is the 4th power of the same prime:

$$N=p^4$$

Using the Unique Factors Trick once more, we discard the base and add one to the exponent to determine that N has 4+1=5 total factors. The correct answer is E.
_________________

"Be challenged at EVERY MOMENT."

“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”

"Each stage of the journey is crucial to attaining new heights of knowledge."

Director
Status: Come! Fall in Love with Learning!
Joined: 05 Jan 2017
Posts: 533
Location: India
Re: For a certain positive integer N, N^3 has exactly 13 unique factors. H  [#permalink]

### Show Tags

20 Mar 2017, 03:24
Bunuel wrote:
For a certain positive integer N, N^3 has exactly 13 unique factors. How many unique factors does N have?

A. 1
B. 2
C. 3
D. 4
E. 5

since 13 is a prime number therefore 12 will be the power of a prime number and that will be N^3
therefore n will be a prime number raised to the power 4
total factor will be 4 +1 =5
_________________

GMAT Mentors

Non-Human User
Joined: 09 Sep 2013
Posts: 9460
Re: For a certain positive integer N, N^3 has exactly 13 unique factors. H  [#permalink]

### Show Tags

25 Mar 2018, 14:32
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: For a certain positive integer N, N^3 has exactly 13 unique factors. H &nbs [#permalink] 25 Mar 2018, 14:32
Display posts from previous: Sort by