2. 92 The largest even number that cannot be expressed as the sum of two odd composite numbers is?

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[Solution] First, we prove: for even numbers not less than 40, they can definitely be expressed as the sum of two odd composite numbers.
Let $k$ be an even number not less than 40.
If the unit digit of $k$ is 2 (i.e., $42,52,62, \cdots$),
then $k=27+5 n(n=3,5,7, \cdots)$;
If the unit digit of $k$ is 4 (i.e., $44,54,64, \cdots$), then $k=9+5 n(n=7,9$, $11, \cdots)$;

If the unit digit of $k$ is 6 (i.e., $46,56,66, \cdots$), then $k=21+5 n(n=5$, $7,9, \cdots)$;

If the unit digit of $k$ is 8 (i.e., $48,58,68, \cdots$), then $k=33+5 n(n=3$, $5,7, \cdots)$;

If the unit digit of $k$ is 0 (i.e., $40,50,60, \cdots$), then $k=35+5 n(n=1$, $3,5, \cdots)$.
Thus, even numbers that cannot be written as the sum of two odd composite numbers must be less than 40.
Next, we verify that 38 is an even number that meets the requirement of the problem.
Since $38=1+37=3+35=5+33=7+31=9+29=11+27$ $=13+25=15+23=17+21=19+19$, 38 cannot be expressed as the sum of two odd composite numbers, hence 38 is an even number that meets the requirement of the problem.