Multiple choice

Which one of the following options is CORRECT given three positive integers $x, y$ and $z$, and a predicate
$$P\left(x\right) = \neg \left(x=1\right)\wedge \forall y \left(\exists z\left(x=y*z\right) \Rightarrow \left(y=x\right) \vee \left(y=1\right) \right)$$

  1. P(x) being true means that x is a prime number.

  2. P(x) being true means that x is a number other than 1.

  3. P(x) is always true irrespective of the value of x.

  4. P(x) being true means that x has exactly two factors other than 1 and x.
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A Correct answer
Explanation

The predicate P(x) states: x ≠ 1 AND for all y, if x = y*z (y divides x), then either y = x OR y = 1. This is precisely the definition of a prime number: a number greater than 1 with no divisors other than 1 and itself. Option A is correct. Option B is incomplete (composites > 1 also satisfy x ≠ 1). Option C is false (false for x = 4, 6, 8, etc.). Option D is wrong (primes have exactly two factors, not 'other than 1 and x').

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