Suppose that $y$ is a positive integer, and $z$ is the largest factor of $y$ such that $z<y$, then let $x=y/z$. Prove that $x$ must be a prime number.
Is there a simple way to solve this? It should be obvious but it's stumping me :/
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Bill Dubuque
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ksj
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2Hint: suppose $x$ is not prime, so $x = ab$ for some $a,b > 1 $. Then think about $z' = az$ – JC574 Aug 11 '14 at 15:44
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1@JC574: Your simple comment is better than any of the answers posted! – TonyK Aug 25 '14 at 17:10
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Hint $\ $ cofactor reflection $\, x \mapsto n/x\,$ is order-reversing on the set of nontrivial factors of $\,n,\,$ therefore it maps a maximal nontrivial factor to a minimal nontrivial factor (necessarily prime, else it would have a smaller factor, contra minimality).
Remark $ $ Extended to common divisors this method yields $\,{\rm lcm}(a,b) = ab/\gcd(a,b)$.
Bill Dubuque
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Let $z$ be the largest factor of $y$ satisfying $y<z$. Suppose $y=xz$ where $x$ is not a prime integer. Then write $x=ab$; $1<a,b <x$; $a,b$ positive integers. But then $az$ also divides $z$ and satisfies $z<az<y$.
Mike
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