Example 7 What is the largest even number that cannot be written as the sum of two odd composite numbers?

The largest even number that cannot be written as the sum of two odd composite numbers is 38. 

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Example 7 What is the largest even number that cannot be written as the sum of two odd composite numbers?
Solution: First, prove that even numbers $n$ greater than 38 can all be written as the sum of two odd composite numbers.

Obviously, if the unit digit of $n$ is 0, then $n=5k+15$; if the unit digit of $n$ is 2, then $n=5k+27$; if the unit digit of $n$ is 4, then $n=5k+9$; if the unit digit of $n$ is 6, then $n=5k+21$; if the unit digit of $n$ is 8, then $n=5k+33$, where $k$ is an integer greater than 2.
Next, find the largest even number that meets the condition.
Among the even numbers less than 40, remove those that can be expressed as the sum of two odd composite numbers.
For numbers with a unit digit of 0, it is easy to verify that only 10 and 20 are qualified.
Similarly, for numbers with a unit digit of 2, 2, 12, 22, and 32 are all qualified; for numbers with a unit digit of 4, only 4 and 14 are qualified; for numbers with a unit digit of 6, only 6, 16, and 26 are qualified; for numbers with a unit digit of 8, 8, 18, 28, and 38 are all qualified.
After elimination, the largest even number less than 40 that meets the condition is 38.
$$
\begin{aligned}
38 & =1+37=3+35=5+33=7+31 \\
& =9+29=11+27=13+25 \\
& =15+23=17+21=19+19
\end{aligned}
$$

None of these are the sum of two odd composite numbers.
None of these are the sum of two odd composite numbers.