If a Positive Integer n has Exactly Two Positive Factors GMAT Data Sufficiency

Question: If a positive integer n has exactly two positive factors, what is the value of n?

1. \(\frac{n}{2}\)is one of the factors of n
2. The lowest common multiple of n and n + 10 is an even number

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:

Approach Solution (1):

Notice that, n has exactly two positive factors simply means that n is a prime number, so its factors are 1 and n itself.

1. \(\frac{n}{2}\) is one of the factors of n. Since \(\frac{n}{2}\) cannot equal to n, then\(\frac{n}{2}=1\)thus n = 2

Sufficient

2. The lowest common multiple of n and n + 10 is an even number. If n is an odd prime, then n + 10 is also odd. the LCM of two odd numbers cannot be even, therefore n is an even prime, so 2

Sufficient

Correct Option: D

Approach Solution (2):

Only prime numbers have two factors
Using S1: \(\frac{n}{2}\) is one of the factors of n

n would be 2
Using S2: The lowest common multiple of n and n + 10 is an even number
LCM of 2 and 12 would be 2

Correct Option: D

Approach Solution (3):

Stem tells us that N = prime (only 2 factors)
S1: \(\frac{n}{2}\) = Only even prime = 2
Therefore n = 2
Sufficient
S2: Test values:
N N + 10
2 12 LCM = 12 Even
3 13 LCM = 39 Odd
5 15 LCM = 75 Odd

We can keep testing, but the constraints of number properties tell us that Even integer = (Odd or Even) * Even, and the only thing that satisfies this is 2

Correct Option: D

“If a positive integer n has exactly two positive factors, what is the value of n?”- 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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