I am trying to prove that for every integer $n \ge 1$, there exists uniquely determined $a > 0$ and $b > 0$ such that $n = a^2 b$, where $b$ is squarefree.
I am trying to prove this using the properties of divisibility and GCD only. Is it possible?
Let me assume that $n = a^2 b = a'^2b'$ where $a \ne a'$ and $b \ne b$'. Can we show a contradiction now?
For existence: given $n\geq1$, there is a maximal-for-divisibility integer $a$ such that $a^2$ divides $n$ and then $n=a^2b$ for some $b$. If $b$ is divisible by the square of a positive integer $c$, then $n$ is divisible by $(ac)^2$, then the maximality of $a$ implies that $ac=a$, that is that $c=1$: this means that $b$ is squarefree.
For uniqueness: suppose now that we also have $n=c^2d$ with $d$ squarefree. The maximality of $a$ implies that $c\mid a$, so there is an $e$ such that $a=ce$, and then from $c^2e^2b=a^2b=c^2d$ we get $e^2b=d$: since $d$ was supposed to be square free, $e=1$. It follows that $a=c$ and $b=d$.
It should be noted that proving there exists a maximal-for-divisibility $a$ such that $a^2\mid n$ depends on the basic properties of the divisibility relation and the GDC. All the rest is more or less tech-free.
For existence, let $a$ be the largest integer, in the usual ordering, such that $a^2$ divides $n$. If $n=a^2q$, then $q$ must be square-free.
For uniqueness, call a positive integer bad if it has two different decompositions $a^2 c$ and $b^2 d$, where $c$ and $d$ are square-free, and $a$ and $b$ are positive. If there are bad positive integers, let $M$ be the smallest bad one.
If $a$ and $b$ are not relatively prime, we can produce a bad positive integer smaller than $M$. So $a$ and $b$ are relatively prime.
We show that $a^2$ and $b^2$ are relatively prime. There are various approaches. One I like is that there exist integers $x$ and $y$ such that $ax+by=1$. Cube both sides. We get $$a^2(ax^3+3x^2by)+b^2(3axy^2+by^3)=1,$$ which says that $a^2$ and $b^2$ are relatively prime.
Since $a^2c=b^2d$ and $a^2$ and $b^2$ are relatively prime, we have $a^2\mid d$. This contradicts the fact that $d$ is square-free, unless $a=1$. Similarly, $b=1$, and therefore $M$ cannot be bad.
Hint $\ $ If $\rm\: a^2 d =n = b^2 c\:$ for squarefree $\rm\:d,c\:$ then $\rm a\:|\:b\:|\:a\:\Rightarrow\:a=b,\:$ since, by your prior question, for $\rm\: z\:$ squarefree, $\rm\ x^2\:|\:y^2 z\:\Rightarrow\: x\:|\:y,\:$ which we apply twice above, in both directions.
