A question regarding the ways of writing even numbers as the sum of two numbers

Question

How can we prove that $\forall n\in\Bbb N$, $\exists a,b:a+b=2n:a,b\not\equiv 0\pmod {2,3}$?

Such a thing, I regard to be definitely true, but thinking about it, I could not find an way of proving such a thing.

Answer 1

$n=2$ is problematic, no?

For $n≥3$ we work by cases:

If $2n=3k$ then $1, 2n-1$ will work.

If $2n=3k+1$ then $5, 2n-5$ will work.

If $2n=3k+2$ then $1, 2n-1$ will work.