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

The answer key for this question suggests looking at $n-15$, $n-25$, $n-35$ to see if we can express n as a sum of two composite odd numbers. However, I don't get where the motivation of express $n$ as such comes from. Does anyone know where the motivation to do this comes from or is there another way to do this?

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  • $\begingroup$ At least $1$ of $n-15$, $n-25$ and $n-35$ will be divisible by $3$ if $n$ is sufficiently big then adding back on either $15, 25$ or $35$ will give you the sum of $2$ composite integers. $\endgroup$ – PJF49 Jun 4 at 18:54
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Since $n-15,\,n-25,\,n-35$ occupy all three residue classes modulo $3$ (in fact they're the earliest such composites differing by $2\times 5=10$), whichever of them is an odd composite multiple of $3$ ensures $n$ admits such a representation, so $n<35+9=44$. Which evens $\lt44$ aren't a sum of any two of the odd composites $9,\,15,\,21,\,25,\,27,\,33,\,35,\,39$? Well, the largest of them would be $38$ since $40=15+25,\,42=15+27$.

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