How many prime factors does x^37 have?

Hi chetan2u,

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

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...

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

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

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

[edit] 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

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..!!!

Hi Bunuel,

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

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