Goldbach Partition - Why Co-Primality?

Question

Any even number $2n$ can be written as the sum of two primes, $p_{a}$ and $p_{b}$. For $n \geq 2$ this is the Goldbach Conjecture.

$$ p_{a} + p_{b} = 2n $$

Why are $p_a$ and $2n$ co-prime? That is, $p_a$ is not a factor of $2n$?

Answer 1

Because $p_a$ and $p_b$ are coprime to each other, they are also both coprime to their sum, $p_a{+}p_b$.

From the contradiction side: clearly if say $p_a$ divides $p_a{+}p_b$, it must also divide $p_b$.

Answer 2

Actually, we can have $p_a = p_b$ not coprime to $2n$. For instance $3 + 3=6$ gives the Goldbach decomposition of $6$.

But if $p_a$ and $p_b$ are distinct they must be coprime, and hence both are coprime to their sum.