Consider a language of predicate logic with a constant sybol $1$, with unary predicate symbols $prime$, $odd$ and $even$, and with a binary predicate symbol $|$.
The intended domain of discourse is the set of non-negative integers, and the intended interpretation of the predicate and constant symbols is what their names suggest. e.g. $prime(x)$ means that $x$ is a prime number, and the binary predicate symbols $|$ represents the "divides" relation, written in infix notation, so $x|y$ means that $y$ can be divided by $x$ without reminder.
Am I correct in my translations of the following sentences into predicate logic with equality, using only the above predicate and constant symbols?
a) One is an odd number.
My attempt: $odd(1)$
b) All prime numbers are odd or even.
My attempt: $\forall x (prime(x) → (odd(x) ∨ even(x)))$
c) No prime number is both odd and even.
My attempt: $ \neg\exists x (prime(x) ∧ (odd(x) ∧ even(x))$
d) Some even numbers can be divided by an odd number without remainder.
My attempt: $ \exists x \exists y (even(x)|odd(y))$
e) Prime numbers can only be divided by one and by themselves without remainder
My attempt: $ \forall x (prime(x) → prime(x)|1 ∨ prime(x)|prime(x)))$
Second attempt at d) $ \exists x \exists y (even(x) ∧ odd(y) ∧ y|x)$
Second attempt at e) $\neg \exists x \exists y (prime(x) ∧ y|x) $ where $y \ne 1$ or $y \ne x$
Fourth attempt at e)
$\neg \exists x \exists y (prime(x) ∧ y|x ∧ y \ne 1 ∧ y \ne x)$