Is positive integer x prime?

Question

(1) x^2 has exactly three factors
(2) x^3 has fewer than five factors

gmat6nplus1 · Answer

1) If  x^2  has exactly 3 factors, the x must be a prime number. Factors of  x^2  would be 1, x, and  x^2 . Pick 2.  x^2 =4. Factors are 1, 2, 4.

2) if you take a prime number, then the total number of factors of  x^3  is 4, but if you take 1, which is positive but not a prime number the number of factors is also smaller than 5. Not sufficient.

Answer A.

Bunuel · Answer

VERITAS PREP OFFICIAL SOLUTION:

Solution: A

Start with Statement (1). Try to find a few numbers that have exactly three factors, such as 4, 9, 25, 49, etc. Notice a pattern? Any number with exactly three factors MUST be a prime number squared. The logic behind this: any integer greater than 1 must have at least two factors, itself and 1. Notice how those factors multiply together to give you the number itself? Any factor of a number must multiply by a different factor of that number directly to give you the number itself, UNLESS that factor is the number’s square root, in which case it multiples by itself to give you the original number without adding another factor. So if an integer has exactly three factors, it has itself, 1, and its square root, which must be prime, since if it weren’t it would add additional factors when squared. SUFFICIENT. Statement (2) isn’t helpful, as x could be 1, which isn’t prime, or 2, which is. (A).

Temurkhon · Answer

St1. x^2 has three total factors means that they are: x, x^2, 1. So x has only two factors, so prime. SUFF

St2. x^3 has < five total factors means that if x was prime they would be: x, x^2, x^3, 1. Should not forget x=1 when we have 1 factor. INSUFF

A

stonecold · Answer

Man i discarded x=1 and hence picked A as the answer..
Lesson learnt => NEVER DISCARD ZERO /ONE from the answer

rhine29388 · Answer

Statement 1 says x^2 has 3 factors which is only possible if x is a prime number and x^2 is prime number squared . It would have 3 factors - 1,x,x^2 So sufficient
Statement 2 says x^3 has fewer than 5 factors . If x = 1 then given integer is prime but the answer to given question is NO. But if we consider x = 2 then answer to given question is YES. so insufficient
Correct Answer - A

ElninoEffect · Answer

What we know: x is a positive number.

(1) x^2 has exactly three factors
Let's check the values of x:
1 = has only one factor. Out.
2 = will have 3 factors.
3 = will have 3 factors.
4 = 5 factors.
In fact for all non prime numbers the number of factors will be more than 3.
 Sufficient.

(2) x^3 has fewer than five factors
It will fit for all prime numbers except in one case which is x=1
x=1 will have 1 factor i.e less than 5 factors and 1 is not a prime number.
 Not sufficient.

A is the answer.

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Is positive integer x prime?

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Is positive integer x prime?

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