# If x is a positive integer, then is x…

## Solution:

We are given that x is a positive integer and must determine whether x is prime.

**Statement One Alone:**

3x + 1 is prime.

Using the information in statement one, x does not necessarily have to be prime. For instance, if x = 2, then 3x + 1 = 7 is prime, or if x = 4, then 3x + 1 = 13 is prime. In the former case, x = 2 is prime; however, in the latter case, x = 4 is not prime. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

**Statement Two Alone:**

5x + 1 is prime.

Using the information in statement two, x does not necessarily have to be prime. For instance, if x = 2, then 5x + 1 = 11 is prime, or if x = 6, then 5x + 1 = 31 is prime. In the former case, x = 2 is prime; however, in the latter case, x = 6 is not prime. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

**Statements One and Two Together:**

Using the information from statements one and two, we still cannot determine whether x is prime. For instance, if x = 2, then both 3x +1 = 7 and 5x + 1 = 11 are prime, or if x = 12, then both 3x + 1 = 37 and 5x + 1 = 61 are prime. In the former case, x = 2 is prime; however, in the latter case, x = 12 is not prime.

**Answer: E**