Example 6. What is the largest even integer that cannot be written as the sum of two odd composite numbers? (If a positive integer can be divided by a positive integer other than 1 and itself, then this positive integer is called a composite number).
(2nd American Invitational Mathematics Examination)
Solution: Because the last digit of a positive even number is $0, 2, 4, 6, 8$.
When the last digit is 0, there are even integers such as $0,10,20,30,40, \cdots$, among which $30=15+15$.

When the last digit is 2, there are even integers such as $2,12,22,32,42,52, \cdots$, among which $42=15+27,52=25+27, \cdots$. Therefore, even integers above 42 do not meet the criteria.

When the last digit is 4, there are even integers such as $4,14,24,34, \cdots$, among which $24=9+15,34=9+25, \cdots$. Therefore, even integers above 24 do not meet the criteria.

When the last digit is 6, there are even integers such as $6,16,26,36,46, \cdots$, among which $36=15+21,46=15+21$, therefore, even integers above 36 do not meet the criteria.

When the last digit is 8, there are even integers such as $8,18,28,38,48,58, \cdots$, among which $48=21+27,58=33+25, \cdots$. Therefore, even integers above 48 do not meet the criteria.

In summary, the largest even integer that cannot be written as the sum of two odd composite numbers can only be 38.