What is the maximum even number that can not be expressed as sum of two composite odd numbers? 
Question: What is the maximum even number that can NOT be expressed as sum of two composite odd numbers?

For example, $14=7+7=5+9=3+11=1+13$ is one of such even numbers, but probably not the maximum number. $24=9+15=3\times3+3\times5$ is not one of such numbers.
I have no idea how to tackle this one. Thanks.
 A: Consider the three odd composites $9,25,35$.  These are, respectively, $0,1,2\pmod 3$.  Thus, if $n$ is an even number, one of $n-9,n-25,n-35$ is an odd composite divisible by $3$  (well, supposing it is $>3$ at least).  Thus $35+3=\fbox {38}$ is the largest even number that might be an example...inspection shows that it  is indeed an example, hence the maximum example.
A: Hint: If $x\geq 18$ is divisible by $6$, then we can write $x$ as $(6n+3)+9$ for some positive integer $n$, where both $6n+3$ and $9$ are composite.
Can you do something similar if $x\equiv 2$ or $4\pmod 6$ and $x$ is large enough?
A: For $n \geq 39$, at least one of the numbers $n - 9, n - 15, n - 21, n-27, n- 33$ (which are all greater than $5$) must be divisible by $5$, hence composite. So you can limit your search to numbers $n \leq 38$. 
As noted in other answers, it turns out that $38$ works. This can be checked as follows. If we had $38 = a + b$ where $a$ and $b$ were odd composite numbers, at least one of them, say $a$, would have to be $\leq 19$. So it's enough to check that $38 - 9 = 29$ and $38 - 15 = 23$ are prime.
