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

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

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New post 17 Nov 2014, 12:29
2
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A
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E

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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.

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

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New post 18 Nov 2014, 02:13
2
Statement 1 tells that the no of factors of x is n+1.
Statement 2 is clearly not sufficient, but it tells that n=2.
Together they tell that x has 3 factors. Answer is C.
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Re: How many factors does x have, if x is a positive integer ?  [#permalink]

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New post 18 Nov 2014, 07:03
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]

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New post 18 Nov 2014, 07:33
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]

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New post 18 Nov 2014, 07:48
1
1
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.
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Re: How many factors does x have, if x is a positive integer ?  [#permalink]

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New post 31 May 2016, 18:00
Bunuel 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?
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Re: How many factors does x have, if x is a positive integer ?  [#permalink]

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New post 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

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

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New post 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]

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