IMO D. As it is “could be” question, plugin x=7, only D holds

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A) is the square of a prime, which can't be prime by definition since it has 3 factors
B) all prime numbers besides 2 are odd, so dividing x by 2 results in a non-int
C) a prime x can only have factors of itself and 1, if x is 3 then 3*3=9, perfect square so again, more than 2 factors
D) subtracting an even from an odd number means that the result is odd, which could be prime (e.g. 7-4=3)
E) any prime squared is already not prime, adding 1 just means it will be even and obviously not a prime number

If x is a prime number greater than 2, which of the following could be a prime number?

All prime numbers are integers.

The only factors of a prime number are that number and 1.

All prime numbers greater than 2 are odd.

Any even number greater than 2 is not prime.

(A) \(x^2\)

The factors of this number are 1, x, and \(x^2\). So this number cannot be prime.

(B) \(\frac{x}{2}\)

Since all prime numbers greater than 2 are odd, \(\frac{x}{2}\) is not an integer, and therefore will not be prime.

(C) \(3x\)

The factors of this number are 1, 3, x, and 3x. So this number cannot be prime.

(D) \(x - 4\)

An odd prime number minus 4 is an odd number. A odd number could be prime.

Example: 11 - 4 = 7

(E) \(x^2 + 1\)

A prime number greater than 2 is odd. The square of an odd number is an odd number. An odd number + 1 is an even number. An even number greater than 2 is not prime.

The correct answer is (D).
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Check the answer choices.....

D) If x = 7, then x - 4 is also prime.
Similarly, if x = 11, then x - 4 is prime.
And if x = 23, then x - 4 is prime.

Answer: D

Aside: Let's see why answer choice E can never be prime.

First of all, let's recognize that if x is a prime number greater than 2, then x is ODD
Any odd number can be written in the form 2k + 1, where k is an integer
For example, 13 = 2(6) + 1, and 37 = 2(18) + 1

E) x² + 1
Replace x with 2k + 1 to get: (2k + 1)² + 1
Expand to get: 4k² + 4k + 1 + 1
Simplify: 4k² + 4k + 2
Factor: 2(2k² + 2k + 1)
This tells us that we can write x² + 1 as a multiple of 2, which means x² + 1 is EVEN, which means it cannot be prime

Cheers,
Brent
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