How many different prime factors does x have?

Question

1)  5x^2  has two different prime factors
2)  x > 1

Source: self-made.

EMPOWERgmatRichC · Answer

Hi Harley1980,

If you're comfortable with the math 'rules' involved, you might be able to answer this question with a 'conceptual' approach. However, TESTing VALUES will help you to get the necessary PROOF of the correct answer.

We're asked for the number of DIFFERENT prime factors of X.

Fact 1: 5X^2 has 2 DIFFERENT prime factors

IF....
X = 2
5X^2 = 20, which has 2 different prime factors (2 and 5)
X = 2, so X has ONE prime factor

IF.....
X = 10
5X^2 = 500, which has 2 different prime factors (2 and 5)
X = 10, so X has TWO different prime factors
Fact 1 is INSUFFICIENT

Fact 2: X > 1

We already have all the work we need to deal with THIS Fact too. The two examples that we used for Fact 1 will 'fit' here....

IF....
X = 2
5X^2 = 20, which has 2 different prime factors (2 and 5)
X = 2, so X has ONE prime factor

IF.....
X = 10
5X^2 = 500, which has 2 different prime factors (2 and 5)
X = 10, so X has TWO different prime factors
Fact 2 is INSUFFICIENT

Combined, we have two different results that already 'fit' BOTH Facts (and provide different results: 1 YES and 1 NO)
Combined, INSUFFICIENT

Final Answer:  E

GMAT assassins aren't born, they're made,
Rich

Harley1980 · Answer

Actually I have planned another trick in this task, but looks like this was unsuccessful attempt and task not as interesting as it was planned.
Great thanks you for your detail analysis.

GSBae · Answer

Were you considering the case that x is not an integer? For example, x^2 could equal 15 (which would make x irrational), so then 5*x^2 = 5*5*3 = 2 prime factors, but x has zero prime factors.

stonecold · Answer

Nice Question  Harley1980 
Key Properties applicable in this question =>
 X and X^n always have the exact same prime factors.

When we multiple a positive integer x with some other positive integer n => The number of prime factors of nx may be greater than or equal to the number of prime factors of x. 
We need the number of prime factors of x.
Statement 1->
5x^2 has primes
x can have 2 primes (if 5 is one of its primes)
x can have 1 prime (if 5 is not one of its primes)
Hence not sufficient.
Statement 2->
x>1
Clearly not sufficient.
Combing the two statements =>
Still not sufficient.

E.g =>
x=3
or 
x=5*3
Satisfy both the equations.

Hence E.

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How many different prime factors does x have?

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How many different prime factors does x have?

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