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

 It is currently 16 Oct 2019, 20:47 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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  S97-01

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 58383

Show Tags 00:00

Difficulty:   55% (hard)

Question Stats: 58% (00:59) correct 42% (00:54) wrong based on 65 sessions

HideShow timer Statistics

How many factors does $$x$$ have, if $$x$$ is a positive integer?

(1) $$x = p^n$$, where $$p$$ is a prime number

(2) $$n^n = n + n$$, where $$n$$ is a positive integer

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 58383

Show Tags

Official Solution:

We cannot easily rephrase the question. Note that we may not need to know $$x$$ in order to know how many factors it has.

Statement (1): INSUFFICIENT. Without knowing the value of $$n$$, we cannot determine the number of factors $$x$$ has.

Statement (2): INSUFFICIENT. This statement by itself is unconnected to the question, because the statement involves only the variable $$n$$, whereas the question only involves the variable $$x$$.

Statements (1) and (2) TOGETHER: SUFFICIENT. First, we should analyze the second statement further, to see whether we can find a unique value of $$n$$.

Since $$n$$ is a positive integer, we can test simple positive integers in an organized fashion, checking for equality of the two sides of the equation.

$$1^1 = 1 + 1$$? No.

$$2^2 = 2 + 2$$? Yes.

$$3^3 = 3 + 3$$? No.

$$4^4 = 4 + 4$$? No.

Notice that the left side of the equation is growing at a much faster rate than the right side, so the equation will not be true for any higher possible values of $$n$$. Thus, we can determine that the value of $$n$$ is 2.

Now, we do not know the value of $$p$$, nor of $$x$$, but we do now know that $$x = p^2$$, with $$p$$ as a prime number. Since a prime number has no factors other than 1 and itself, we can see that $$x$$ has no factors other than 1, $$p$$, and $$p^2$$. Thus, $$x$$ has exactly 3 factors, and we can answer the question definitively.

_________________
Intern  B
Joined: 23 May 2017
Posts: 3

Show Tags

Combining statements 1 and 2, for all other prime numbers other than 1, the factors are 1, p, and p2. what if the prime number is 1, it has only 1 factor. so how is it definitive ? I would say both statements together are insufficient. Can you please explain ?

thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 58383

Show Tags

Ganeshsrinivasan wrote:
Combining statements 1 and 2, for all other prime numbers other than 1, the factors are 1, p, and p2. what if the prime number is 1, it has only 1 factor. so how is it definitive ? I would say both statements together are insufficient. Can you please explain ?

thanks

1 is NOT a prime number.

A Prime number is a positive integer with exactly two distinct positive divisors: 1 and itself. The smallest prime (and the only even prime) is therefore 2.

For more on Number Theory check the following post: https://gmatclub.com/forum/math-number- ... 88376.html
_________________
Intern  Joined: 30 Jan 2019
Posts: 1

Show Tags

What about 1 for a prime number? that only has 1 factor so you can not tell if it will have 3 or 1 factors. Not sufficient
Math Expert V
Joined: 02 Sep 2009
Posts: 58383

Show Tags

jtwill5 wrote:
What about 1 for a prime number? that only has 1 factor so you can not tell if it will have 3 or 1 factors. Not sufficient

Please check the post just above yours: https://gmatclub.com/forum/s97-184689.html#p1880964
_________________ Re: S97-01   [#permalink] 01 Feb 2019, 00:14
Display posts from previous: Sort by

S97-01

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

Moderators: chetan2u, Bunuel

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne  