How many factors does x have, if x is a positive integer ? : Data Sufficiency (DS)

2014-11-17T11:29:51-08:00

How many factors does [m]x[/m] have, if [m]x[/m] is a positive integer? (1) [m]x = p^n[/m], where [m]p[/m] is a prime number (2) [m]n^n = n + n[/m], where [m]n[/m] is a positive integer

akumar5

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

18 Nov 2014, 07:03

1

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x is a positive integer

from statement 1 : x=p^n, where p is prime
so the no of factors will be n+1.
value of n will lead to complete solution.

from statement 2: n^n=n+n, where n is positive integer
n^n=2n
n^n-2n=0
n(((n^(n-1))-2)=0
hence either n=0 or n^(n-1)=2 that is n=2

combining both statements will yield two different no of factors for two different value of n.

Answer E

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

18 Nov 2014, 07:33

1

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We are told that n is +ve, you did everything correct but n cant be 0 as n is +ve. So n=2.

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

31 May 2016, 18:00

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Bunuel wrote:

Tough and Tricky questions: Divisibility/Multiples/Factors.

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

Kudos for a correct solution.

Official Solution:

How many factors does \(x\) have, if \(x\) is a positive integer?

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.

Answer: C.

@Bunel, if we know P is prime, and all prime numbers have only three factors, and exponents do not produce new factors, why cant we say that # factors for N = 3 since N itself the just a prime number to a power?

B

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

31 May 2016, 18:39

akumar5 wrote:

x is a positive integer

from statement 1 : x=p^n, where p is prime
so the no of factors will be n+1.
value of n will lead to complete solution.

from statement 2: n^n=n+n, where n is positive integer
n^n=2n
n^n-2n=0
n(((n^(n-1))-2)=0
hence either n=0 or n^(n-1)=2 that is n=2

combining both statements will yield two different no of factors for two different value of n.

Answer E

It said n is a positive integer, so n=|=0
Ans-C

Sent from my XT1033 using GMAT Club Forum mobile app

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

24 Aug 2016, 06:55

Such an amazing Question
Here is y approach =>
Here we need to get the number of factors of x for x >0 and it is an integer
Statement 1 -> x=prime ^n okay if n=2=> factors =3 ;
n=4 => factors =5
Rule => if X= A^a*B^b where A and B are PRIME=> Number of factors of = (a+1)*(b+1)
Hence Insuff
Statement 2 => here only value possible is n=2
NOTE => 2 is a funny number =>
Its the only even prime
only number for which the square is twice the number
Its one of the two numbers for which number of divisors = number itself (other being 1)
2,3 are the only two consecutive numbers that are prime
and much more...
here though we have no clue on x => insuff
Combining them => x=Prime ^2 => number of factors =(2+1)=> 3
Smash that C

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

04 Apr 2019, 08:08

n^n equals 2n only for n=2; substituting in 1st options gives us a definite ans.

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

05 Feb 2020, 06:26

rjivani wrote:

Bunuel wrote:

Tough and Tricky questions: Divisibility/Multiples/Factors.

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

Kudos for a correct solution.

Official Solution:

How many factors does \(x\) have, if \(x\) is a positive integer?

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.

Answer: C.

@Bunel, if we know P is prime, and all prime numbers have only three factors, and exponents do not produce new factors, why cant we say that # factors for N = 3 since N itself the just a prime number to a power?

Can we please have an answer to this?

L

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

05 Feb 2020, 06:44

Expert Reply

vikrantkumar13 wrote:

rjivani wrote:

Bunuel wrote:

Tough and Tricky questions: Divisibility/Multiples/Factors.

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

Official Solution:

How many factors does \(x\) have, if \(x\) is a positive integer?

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.

Answer: C.

@Bunel, if we know P is prime, and all prime numbers have only three factors, and exponents do not produce new factors, why cant we say that # factors for N = 3 since N itself the just a prime number to a power?

Can we please have an answer to this?

The question itself is not right.

For one, primes have two factors not three: 1 and itself. Next, exponentiation does not "produce" new PRIME FACTORS but it surely "produces" more factors. For example, 2 has two factors 1 and 2 but 2^2 = 4 has three factors 1, 2, and 4 (but the same PRIME factor 2).

L

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

05 Feb 2020, 09:00

Bunuel wrote:

Tough and Tricky questions: Divisibility/Multiples/Factors.

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

Kudos for a correct solution.

Asked: How many factors does \(x\) have, if \(x\) is a positive integer?

(1) \(x = p^n\), where \(p\) is a prime number
Since n is unknown
NOT SUFFICIENT

(2) \(n^n = n + n\), where \(n\) is a positive integer
No relation between x & n is established.
NOT SUFFICIENT

(1) + (2)
(2) \(n^n = n + n\), where \(n\) is a positive integer
n = 2
(1) \(x = p^n\), where \(p\) is a prime number
x = p^2
x has 3 factors since p is a prime number
SUFFICIENT

IMO C

L

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

06 Feb 2020, 00:59

Bunuel wrote:

Tough and Tricky questions: Divisibility/Multiples/Factors.

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

Kudos for a correct solution.

#1\(x =p^n\), where \(p\) is a prime number
n value not know; as n=1 ; factors ox x = 2 else when n=2 then factors of x = 3 insufficient
#2
) \(n^n = n + n\), where \(n\) is a positive integer
no info about p ,x ; insufficient
from 1 &2
n= 2 and factos square of prime no ; 3
IMO C; sufficient

B

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

27 Sep 2020, 01:53

Hi Bunuel @VeritasPrepKarishma: should the question stem be "positive" factors because we get only positive factors by n+1. Also Is it that the total number of positive factors = the total number of negative factors For eg. 2^2 positive factors = 3 positive factors (2+1) but there are 3 negative factors too. so 2(n+1) for total number of factors (positive and negative).

Thanks in advance.

L

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

28 Sep 2020, 03:07

Expert Reply

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

DS question with 1 variable: Let the original condition in a DS question contain 1 variable. Now, 1 variable would generally require 1 equation for us to be able to solve for the value of the variable.We know that each condition would usually give us an equation and Since we need 1 equation to match the numbers of variables and equations in the original condition,the logical answer is D. The answer could be A, B or D, but the default answer will be D.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find number of factors of 'x' - where 'x' is a positive integer.

Second and the third step of Variable Approach: From the original condition, we have 1 variable (x).To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Let’s take a look at each condition.

Condition(1) tells us that x = \(p^n\) - where 'p' is a prime number.

=> number of factors will be (n + 1), but as 'n' is unknown, we can have many values

Since the answer is not unique , condition(1) is not sufficient by CMT 2.

Condition(2) tells us that \(n^n\) = n + n .

=> 'x' is unknown hence cannot tell about its number of factors.

Since the answer is not unique , condition(2) is not sufficient by CMT 2.

Let’s take a look at both conditions together.

=> from condition (2) \(n^n\) = n + n only possible when n = 2 and condition(1) x = \(p^2\)

=> Number of factors= 2 + 1 = 3

Since the answer is unique , both conditions together are sufficient by CMT 2.

So, C is the correct answer.

Answer: C

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

22 Mar 2022, 09:56

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Re: How many factors does x have, if x is a positive integer ? [#permalink]

22 Mar 2022, 09:56

GMAT Data Sufficiency (DS) Questions

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