How many prime factors does x^37 have?

Question

How many prime factors does  x^{37}  have?

(1) The number of prime factors of 16x is one more than the number of prime factors of  x^4

(2)  2x^{16}  has 2 prime factors

chetan2u · Answer

ans C....

1) statement 1 tells us that 16x has one more prime factor than  x^4 , which will be same as for x.... therefore it tells us that x is not multiple of 2... insufficient...

2) statement 2 tells us that  2x^{16}  has two prime factor .... it tells us that that one prime factor is 2 along with one more prime factor...
does not tell us that x also has a 2 in it... insufficient...

combined it tells us that x has only one prime factor that is itself.... sufficient

TARGET730 · Answer

Hi chetan2u,

Can you please explain "combined it tells us that x has only one prime factor that is itself." ?

Bunuel · Answer

How many prime factors does  x^{37}  have?

Real GMAT question would mention that x is a positive integer. ALL GMAT divisiblity questions are limited to positive integers. So, I'd assume that it's given that x is a positive integer.

First of all note that exponentiation does not "produce" primes, meaning that for positive integers m and n, m^n would have as many prime factors as m itself does.

(1) The number of prime factors of 16x is one more than the number of prime factors of x^4 --> 16x =  2 ^4*x. So it's given that 2^4*x has one more prime than x^4, which means that 2 is NOT a prime factor of x. We know nothing else about x. Not sufficient.

(2)  2x^{16}  has 2 prime factors --> this implies that x itself can have 1 prime (something other than 2) or 2 primes (2 and something other than 2). Not sufficient.

(1)+(2) Since from (1) we know that 2 is NOT a factor of x, then from (2) it follows that x must have only 1 prime. Thus x^37 also will have only 1 prime. Sufficient.

Answer: C.

Hope it's clear.

chetan2u · Answer

hi  TARGET730

1) statement 1 tells us that 16x has one more prime factor than x^4, which will be same as for x.... therefore it tells us that x is not multiple of 2... insufficient...
 the power will have same prime factors as x itself 
2) statement 2 tells us that 2x^{16} has two prime factor .... it tells us that that one prime factor is 2 along with one more prime factor...
does not tell us that x also has a 2 in it... insufficient...
 this tells us that 2x^{16} has two factors.. one is two and other could be anything.... but x may be multiple of two then x will have 2 prime factors... if x is not multiple of 2 then x will have only one prime factor...

combined it tells us that x has only one prime factor that is itself.... sufficient
 now 2 is not in x so x contains one prime factor which can be itself or some root of x...

ak1802 · Answer

Here's my explanation, I tried to keep it simple:

The question is essentially asking us if x is a prime, or can be factored into a single prime.

St (1): tells us that 2 is not a factor of x. We know this because if 16(x) has one less prime factor than x, that "one less prime factor" is 2. However, we still do not know whether x is prime or not. N.S.
2^8 =16, for those who wonder how that association is made.

For example:

x can equal 15 which satisfies the condition in st(1), but doesn't solve the problem.

x can also equal 3 which satisfies the condition in st(1) and DOES solve the problem.

St (2) tells us that 2(x)^16 only has two primes. This means that x can be factored into a single prime number OR x has 2 and another prime as its prime factors.

St(1) and St (2) tell us that, working with St(2) first; 2(x)^16 only has two primes. St(1) tells us that 2 is not a factor of x. Therefore x  must be a prime number  must be a prime number or an integer that can be factored into a single prime number, other than 2.

C.

Posted from my mobile device

chetan2u · Answer

hi, 
it is not necessary x will be a prime number ... it can be a square of a prime number... only surety is that it has only one prime number as factor

ak1802 · Answer

Yes, you're correct. I stated above that the question is asking if X can either be prime, or factored into a single prime.

didn't notice the red text on my phone. You're right, error on my part in St(1) and St(2). X must be an integer, that can be factored into a  single prime number , other than 2.

Posted from my mobile device

stonecold · Answer

Such an Amazing Question.
Here is my Approach ->

Wee need the number of prime factors of  X^{37} 
 RULE -> Raising a positive integer to a positive exponent does not change its prime factors. 
Hence  X^{37}  will have the exact same prime factors as  X .
Thus in a way the Question is asking for the =>  number of prime factors of  X .

Lets Dive into statements

Statement 1-->  
 16X => one more prime factor than  X^4 
Again X^4 will have the exact same prime factors as  X .
So  16X  has one more prime than X
 16X=2^4*X  => That additional Prime must be 2.
Thus  X  does have 2 as its Prime factor.
But we don't know the number of Prime factors of X.
Hence Not sufficient.

Statement 2-->  
 2X^{16}  has two Prime factors.
This means that  X  can have either one prime factor (exclusive of 2)
Or it can have Two prime factors(inclusive of 2)
Hence not sufficient.

Combing the two statements-->  
We can say that X does have two as its prime=> 2*X^{16}  has two primes =>  X  must have only one prime.
Hence Sufficient

Hence C

Great Question..!!!

MI83 · Answer

Hi Bunuel,

why the statement (2) cannot allow us to conclude that X has only one prime factor definitely other than 2?

bumpbot · Answer

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How many prime factors does x^37 have?

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