# Determine the largest even positive integer [closed]

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

• This is purely a math question, there is no puzzle, no riddle, no nothing. Belongs on Math.SE – Joe Nov 7 '14 at 13:13
• @Joe, it is a puzzle to evaluate the answer, the answer is too trivial to belong on maths.SE – Kenshin Nov 7 '14 at 13:14
• 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 :-) – Joe Nov 7 '14 at 13:16
• 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. :) – Chris Nov 7 '14 at 14:50

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.