Cofactor reflection maps greatest proper divisor to least (prime) proper divisor

Question

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 :/

Answer 1

Hint: Use the fundamental theorem of arithmetic.

Answer 2

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 $\,{\rm lcm}(a,b) = ab/\gcd(a,b)$.

Answer 3

Hint If $d |x$ then $dz |y$ and $z \leq dz \leq y$. What can $dz$ be?

Answer 4

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 $z<az<y$.