<s>[INST] If $n>3$ is not prime, show that we can find positive integers $a,b,c$ such that $n=ab+bc+ca+1$. 
If $n>3$ is not prime, show that we can find positive integers $a,b,c$ such that $n=ab+bc+ca+1$.

When $n=4$, pick $a=b=c=1$. When $n=6,$ pick $a=2$ and $b=c=1$. When $n=8$, pick $a=b=c=2$. I don't get the pattern here. I know that if $n>3$ is not prime, then $n=xy,$ where $1<x<n$ and $1<y<n$. But how do I proceed from here? 
 [/INST] Notice that $xy = (x-1)(y-1)+(y-1) + (x-1)+1.$ Choose $a=x-1,b=y-1, $ and $c=1$. All are positive as $x,y>1$.
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