Question: If x is a positive integer greater than 1, is x a prime number?

1. x does not have a factor p such that 2 < p < x
2. The product of any two factors of x is greater than 2 but less than 10

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Solution with Explanation:
Approach Solution (1):

(1) x does not have a factor p such that 2 < p < x. Notice that all odd primes satisfy this statement as well as integer 4 (4 does not have a factor p such that 2 < p < 4)
Not sufficient

(2) The product of any two factors of x is greater than 2 but less than 10. This implies that x can be 3, 5, and 7
Sufficient

Notice that x cannot be an even number because any even number has 1 and 2 as its factors and the product of these factors is 2, not greater than 2 as given in the statement. Also, notice that x cannot be 9 because 3 and 9 both are factors 9 and 3 * 9 = 27 > 10

Correct Option: B

Approach Solution (2):

S1 says that if you prime-factorize x, it won’t have a prime which is greater than 2 but less than x itself. Prime numbers generally don’t have factors which are greater than 1 and less than this prime itself (from definition of primes: a prime number is a positive integer with exactly two factors 1 and itself). So, any odd prime will satisfy this condition. But a not only odd prime, 4 also does not have a prime which is greater than 2 but less than 4 itself (factors of 4 are 1, 2, and 4). So, odd primes as well as 4 satisfy (1), which makes this statement not sufficient

Correct Option: B

Approach Solution (3):

(1) x does not have a factor p such that 2 < p < x. Notice that all odd primes satisfy this statement as well as integer 4 (4 does not have a factor p such that 2 < p < 4)
Not sufficient

(2) This talks of any two factors, but you are just looking at two factors here…
So if you are looking at 9…
Factors are 1, 3, 9…
So if you pick 3 and 9, the product is 3 * 9 = 27
1 * 3 and 1 * 9 may be less than 10 but we are looking at any two
Sufficient

Correct Option: B

“If x is a positive integer greater than 1, is x a prime number?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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