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- Feb 14, 2012

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- Thread starter anemone
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- Thread starter
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- #1

- Feb 14, 2012

- 3,812

- Thread starter
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- #2

- Feb 14, 2012

- 3,812

$\begin{align*}f(x)&=(x+a)(x+b)(x+c)-(x-d)(x-e)(x-f)\\&=Sx^2+(ab+bc+ca-de-ef-fd)x+(abc+def)\end{align*}$

are multiples of $S$. Evaluating $f$ at $d$, we get that $f(d)=(a+d)(b+d)(c+d)$ is a multiple of $S$.

So this implies that $S$ is composite, since $a+d,\,b+d,\,c+d$ are all strictly less than $S$.