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

Question

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

Bunuel · Accepted Answer

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.

sunaimshadmani · Answer

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.

akumar5 · Answer

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

sunaimshadmani · Answer

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

rjivani · Answer

@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?

jaydev2050 · Answer

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

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stonecold · Answer

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

prabhasis · Answer

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

vikrantkumar13 · Answer

Can we please have an answer to this?

Bunuel · Answer

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).

Kinshook · Answer

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

Archit3110 · Answer

#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

kagrawal16 · Answer

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.

Thanks in advance.

MathRevolution · Answer

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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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 ?

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