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The book is trying to prove the fundamental theorem of arithmetic but does he mean by saying that at least one of the inequality must be true with strict inequality ? And from where did he come up with this last inequality ? enter image description here

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    $\begingroup$ They cannot both be equalities, since otherwise $p^2=n=p'^2$ which is impossible with $p\neq p'$. Now we have, say $p\leq\sqrt{n}, p'<\sqrt{n}$, so $pp'<n$. $\endgroup$ – Wojowu Apr 17 at 22:03
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Since $p\ne p'$, you cannot have both $n=p^2$ and $n=p'^2$. Thus, say, $n>p'^2$.

If we multiply the inequalities, we get $n^2>p^2p'^2$, so $n>pp'$.

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