If x is a positive integer, then is x prime? 3x + 1 is prime 5x + 1 is prime...

Understanding the Question

We're asked: "If x is a positive integer, then is x prime?"

This is a yes/no question. We need to determine whether x is prime or not prime.

What "Sufficient" Means Here

For a statement to be sufficient, it must allow us to definitively answer either:

• "YES, x is prime" for all valid values of x, OR
• "NO, x is not prime" for all valid values of x

If different valid values of x lead to different answers (some prime, some not prime), then we have insufficient information.

Key Insight

Since we're dealing with conditions that make linear expressions prime (\(3\mathrm{x} + 1\) and \(5\mathrm{x} + 1\)), the most efficient approach is to test small positive integer values of x. Finding even one prime x and one non-prime x that satisfy a condition immediately proves that condition is insufficient.

Analyzing Statement 1

Statement 1 tells us: \(3\mathrm{x} + 1\) is prime

Let's test values systematically to see which positive integers x could work:

• If \(\mathrm{x} = 1\): \(3(1) + 1 = 4 = 2 \times 2\) → NOT prime ✗
• If \(\mathrm{x} = 2\): \(3(2) + 1 = 7\) → IS prime ✓
• If \(\mathrm{x} = 4\): \(3(4) + 1 = 13\) → IS prime ✓
• If \(\mathrm{x} = 6\): \(3(6) + 1 = 19\) → IS prime ✓

Pattern Note: For \(3\mathrm{x} + 1\) to be prime (and greater than 2), it must be odd. This means \(3\mathrm{x}\) must be even, so x must be even.

Now let's check if these valid x values are prime:

• \(\mathrm{x} = 2\) → IS prime
• \(\mathrm{x} = 4 = 2 \times 2\) → NOT prime
• \(\mathrm{x} = 6 = 2 \times 3\) → NOT prime

Since we get different answers (YES for \(\mathrm{x} = 2\), NO for \(\mathrm{x} = 4\)), Statement 1 alone is NOT sufficient.

[STOP - Statement 1 is Not Sufficient!]

This eliminates choices A and D.

Analyzing Statement 2

Now we forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: \(5\mathrm{x} + 1\) is prime

Testing values of x:

• If \(\mathrm{x} = 1\): \(5(1) + 1 = 6 = 2 \times 3\) → NOT prime ✗
• If \(\mathrm{x} = 2\): \(5(2) + 1 = 11\) → IS prime ✓
• If \(\mathrm{x} = 3\): \(5(3) + 1 = 16 = 2^4\) → NOT prime ✗
• If \(\mathrm{x} = 4\): \(5(4) + 1 = 21 = 3 \times 7\) → NOT prime ✗
• If \(\mathrm{x} = 5\): \(5(5) + 1 = 26 = 2 \times 13\) → NOT prime ✗
• If \(\mathrm{x} = 6\): \(5(6) + 1 = 31\) → IS prime ✓

Valid values include \(\mathrm{x} = 2\) and \(\mathrm{x} = 6\).

Checking if these x values are prime:

• \(\mathrm{x} = 2\) → IS prime
• \(\mathrm{x} = 6 = 2 \times 3\) → NOT prime

Again, we get different answers (YES for \(\mathrm{x} = 2\), NO for \(\mathrm{x} = 6\)), so Statement 2 alone is NOT sufficient.

[STOP - Statement 2 is Not Sufficient!]

This eliminates choice B.

Combining Both Statements

Since we've eliminated A, B, and D, we need to check if both statements together are sufficient.

Using BOTH conditions:

• \(3\mathrm{x} + 1\) is prime AND
• \(5\mathrm{x} + 1\) is prime

We need x values that satisfy both conditions simultaneously.

From our previous analysis:

• \(\mathrm{x} = 2\):
  • \(3(2) + 1 = 7\) (prime) ✓
  • \(5(2) + 1 = 11\) (prime) ✓
  • Both conditions satisfied!

• \(\mathrm{x} = 4\):
  • \(3(4) + 1 = 13\) (prime) ✓
  • \(5(4) + 1 = 21 = 3 \times 7\) (not prime) ✗
  • Does not satisfy both conditions

• \(\mathrm{x} = 6\):
  • \(3(6) + 1 = 19\) (prime) ✓
  • \(5(6) + 1 = 31\) (prime) ✓
  • Both conditions satisfied!

So \(\mathrm{x} = 2\) and \(\mathrm{x} = 6\) both satisfy both conditions.

The crucial observation:

• \(\mathrm{x} = 2\) is prime
• \(\mathrm{x} = 6\) is NOT prime

Even with both statements combined, we still get different answers to our question "Is x prime?". Therefore, both statements together are NOT sufficient.

[STOP - Combined Statements are Not Sufficient!]

The Answer: E

The statements together are not sufficient because we found valid values of x (namely \(\mathrm{x} = 2\) and \(\mathrm{x} = 6\)) that satisfy both conditions but give different answers to whether x is prime.

Answer Choice E: "Statements (1) and (2) TOGETHER are NOT sufficient."