If x is a positive integer, is x prime?
(1) x has the same number of factors as y2, where y is a positive integer greater than 2.
(2) x has the same number of factors as z, where z is a positive integer greater than 2.
Is there an error in the explanation for Statement (1)? It appears so to me.
Could a MGMAT instructor please so kindly verify?
Thanks!
dataiwandude
GMAT Forum
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- dataiwandude
- dataiwandude
Re: If x is a positive integer, is x prime?
I did not post the answer and explanation. Here it is:
The question stem tells us that x is a positive integer. Then we are asked whether x is prime; it is helpful to remember that all prime numbers have exactly two factors. Since we cannot rephrase the question, we must go straight to the statements.
(1) SUFFICIENT: If x has the same number of factors as y2, then x cannot be prime. A prime number is a number that has only itself and 1 as factors. But a square has at least 3 prime factors. For example, if y is prime, y = 2, then y2 = 4, which has 1, 2, and 4 as factors. If the root (in this case y) is not prime, then the square will have more than 3 factors. For example, if y = 4, then y2 = 16, which has 1, 2, 4, 8, and 16 as factors. In either case, x will have at least 3 factors, establishing it as nonprime.
(2) INSUFFICIENT: If z is prime, then x will have only two factors, making it prime. But if z is nonprime, it will have either one (if z = 1) or more than two factors, which means x will have either one or more than two factors, making x nonprime. Since we do not know which case we have, we cannot tell whether x is prime.
The question stem tells us that x is a positive integer. Then we are asked whether x is prime; it is helpful to remember that all prime numbers have exactly two factors. Since we cannot rephrase the question, we must go straight to the statements.
(1) SUFFICIENT: If x has the same number of factors as y2, then x cannot be prime. A prime number is a number that has only itself and 1 as factors. But a square has at least 3 prime factors. For example, if y is prime, y = 2, then y2 = 4, which has 1, 2, and 4 as factors. If the root (in this case y) is not prime, then the square will have more than 3 factors. For example, if y = 4, then y2 = 16, which has 1, 2, 4, 8, and 16 as factors. In either case, x will have at least 3 factors, establishing it as nonprime.
(2) INSUFFICIENT: If z is prime, then x will have only two factors, making it prime. But if z is nonprime, it will have either one (if z = 1) or more than two factors, which means x will have either one or more than two factors, making x nonprime. Since we do not know which case we have, we cannot tell whether x is prime.
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