• Check a few numbers by hand to get a rough idea.
• Divide all other numbers into three cases in terms of modulo $6$.

The desired integers are ${2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 26, 28, 32, 38 }$. It can be checked that these cannot be expressed as the sum of two composite odd numbers. We prove that any other even positive integer $n$ can be expressed as the sum of two composite numbers. All the other even positive integers can be divided into three cases:

Case 1: $n=6k$ and $n\geq18$. $n$ can be written as $n=(n-9)+9$. Here, $n-9\geq9$ and $3 \mid n-9$.

Case 2: $n=6k+2$ and $n\geq44$. $n$ can be written as $n=(n-35)+35$. Here, $n-35\geq9$ and $3 \mid n-35$.

Case 3: $n=6k+4$ and $n\geq34$. $n$ can be written as $n=(n-25)+25$. Here, $n-25\geq9$ and $3 \mid n-25$.