SMO 2012 Question 7

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?

• 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. – PJF49 Jun 4 at 18:54

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 $$40=15+25,\,42=15+27$$.