induction - Prove that every natural number $n >6$ can be written as a sum $a+b$ where $a, b∈mathbb Nsetminus{1}$ and$ gcd(a, b) = 1$

Is this the correct approach to this problem:
I tried induction.
This is what the professor said: Hint: Treat the cases $n$ even and odd separately.

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Inductive hypothesis:

$n=a+b$ for $n>6$

Base case

Let $n = 7$. Then $3+4=7$ where $3,4$ have a gccd of $1$.

Induction Step

Induction Step:
Prove: $n+1=a+b$

Case 1: Let $a$ be $a + 1$. This is possible because $a$ is in the natural numbers.
So $a+b+1=n+1$. By the inductive hypothesis $n=a+b$ so we can substitute it. Therefore $n+1=n+1$.

Case 2: Let $b$ be $b+1$.
This is symmetrical to case one.