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Determine the largest even positive integer that cannot be written as the sum of two odd composite positive integers.

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closed as off-topic by Joe, skv, wbogacz, generalcrispy, JonTheMon Nov 7 '14 at 14:02

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    $\begingroup$ This is purely a math question, there is no puzzle, no riddle, no nothing. Belongs on Math.SE $\endgroup$ – Joe Nov 7 '14 at 13:13
  • $\begingroup$ @Joe, it is a puzzle to evaluate the answer, the answer is too trivial to belong on maths.SE $\endgroup$ – Kenshin Nov 7 '14 at 13:14
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    $\begingroup$ I disagree, since you could argue that any question is "a puzzle to evaluate the answer". Just my opinion though, and I won't get into a slagging match over it :-) $\endgroup$ – Joe Nov 7 '14 at 13:16
  • $\begingroup$ Personally if I were to take issue with the question is that it doesn't define what a composite number is. I'm a mathematician and I had to go look it up. :) $\endgroup$ – Chris Nov 7 '14 at 14:50
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Considering the remainder after dividing by 3, the smallest odd composite number leaving a remainder of 1 is 25 and the smallest one leaving a remainder of 2 is 35.

For any even positive integer greater than or equal to 40:

If it's a multiple of 3, then it's 9 plus an odd multiple of 3;

If it's one more than a multiple of 3, then it's 25 plus an odd multiple of 3;

If it's two more than a multiple of 3, then it's 35 plus an odd multiple of 3.

Therefore the largest possible one is 38. This has a remainder of 2 when divided by 3, which can only be obtained by 1+1 or 0+2. However, 25+25 and 9+35 (the smallest numbers with those remainders) are both too large. Therefore there is no way to express 38 as the sum of two odd composite positive integers.

38 is the answer.

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