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I've been trying to understand the proof for a solution to the Euler 9 problem. I'm on this site under the heading "Solving the problem". I've understood the parts that came before it (excluding the "proof" )

[IMO, reading the project euler problem is not necessary to understand what's happening below]

enter image description here

I'm specifically wondering how the author concluded that since $m|(s/2)\implies m<\sqrt{s/2}$. I mean, even $12$ divides $48$ but $12>\sqrt(48)\approx 7$.

What am I missing here?

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    $\begingroup$ $s=2m(m+n)d \implies s/2=m(m+n)d>m^2$ $\endgroup$ – Joffan Aug 16 '17 at 14:06
  • $\begingroup$ @Joffan Ok, i get this, and I suppose this has nothing to do whether $m|(s/2)$. Am I correct? $\endgroup$ – Gaurang Tandon Aug 16 '17 at 14:08
  • $\begingroup$ It's implied by the same equation, since $m$ is a simple factor $\endgroup$ – Joffan Aug 16 '17 at 14:09
  • $\begingroup$ @Joffan No, I meant that in case had the equation been $s/2=(m+1)(m+n)d$, the result that $m<\sqrt{s/2}$ would be unaffected, even though now $m$ does not divide $s/2$. $\endgroup$ – Gaurang Tandon Aug 16 '17 at 14:11
  • $\begingroup$ @Joffan that's what I needed "Yes, they are two outcomes of the given equation, not derived from each other (in either direction" Please post an answer. Thanks! $\endgroup$ – Gaurang Tandon Aug 16 '17 at 14:14
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$s=2m(m+n)d \implies s/2=m(m+n)d>m^2$ so $\color{blue}{m<\sqrt {s/2}}$

Also, $s/2=m(m+n)d \implies \color{blue}{m \mid s/2}$

So those are two outcomes of the given equation, not derived from each other (in either direction).

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