7. Determine the largest even positive integer which cannot be expressed as the sum of two composite odd positive integers.
7. Answer: 38
Solution. Let $n$ be an even positive integer. Then each of the following expresses $n$ as the sum of two odd integers: $n=(n-15)+15(n-25)+25$ or $(n-35)+35$. Note that at least one of $n-15 n-25 n-35$ is divisible by 3 so that it is composite, and henoe $n$ can be expressed as the sum of two composite odd numbers if $n>38$. Indeed, it can be verified that 38 cannot be expressed as the sum of two composite odd positive integers.