Sum of Two Odd Composite Numbers

What is the largest even integer that cannot be written as the sum of two odd composite numbers? Source: AIME 1984, problem 14.

Note

$24 = 3\cdot 3 + 3\cdot 5$ , and so if

$2k$ has a representation as the sum of even multiples of 3 and 5, say

$2k = e_3\cdot 3 + e_5\cdot 5$ , we get a representation of

$2k+24$ as a sum of odd composites via

$2k+24 = (3+e_3)\cdot 3 + (5+e_5)\cdot 5$ . But by the Frobenius coin problem every number

$k > 3\cdot 5 -3-5 = 7$ has such a representation, hence every number

$2k > 14$ has a representation as the sum of even multiples of 3 and 5. Thus every number

$n > 24+14=38$ has a representation as the sum of odd composites. Checking, we see that

$\boxed{38}$ has no representation as a sum of odd composites.

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