Application of Unique Factorisation Theorem in Proof

Question

CONTEXT: Proof question made up by uni math lecturer

Suppose you have $x+y=2z$ (where $x$ and $y$ are consecutive odd primes) for some integer $z>1$, and that you need to prove that $x+y$ has at least three prime divisors (that don't have to be distinct).

Is it sufficient to say that, according to the unique factorisation theorem for integers, since $z$ can be expressed as a product of primes (and we already have the factor of $2$ which is prime), we know that $x+y$ has at least three prime divisors?

Or, would you need to do further working to show that $z=ab$ for some primes $a$ and $b$?

Answer 1

$z$ is strictly between $x$ and $y$, hence, not a prime.

Answer 2

Is it sufficient to say that, according to the unique factorisation theorem for integers, since $z$ can be expressed as a product of primes (and we already have the factor of $2$ which is prime), we know that $x+y$ has at least three prime divisors?

No, that's not enough. In fact, how do you know $z$ is not prime?

To prove $z$ is not prime, consider the fact that $y=x+2$, and therefore, $2z=x+x+2=2(x+1)$. Now, since $z=x+1$, can you conclude that $z$ is not prime?