Proving $\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$ How would you go about proving that $$\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$$

for any two integers $a$ and $b$?

Intuitively it is true because when you divide $a$ and $b$ by $\gcd(a,b)$ you cancel out any common factors between them resulting in them becoming coprime. However, how would you prove this rigorously and mathematically?
Let $d=\gcd(a,b)$. Let $a=md$ and $b=nd$. If some $k\gt 1$ divides $m$ and $n$, then $kd$ divides $a$ and $kd$ divides $b$, contradicting the fact that $d$ is the greatest common divisor of $a$ and $b$.
Suppose $d \mid \dfrac a{\gcd(a,b)}$ and $d\mid \dfrac b{\gcd(a,b)}$.

Then $nd = \dfrac a {\gcd(a,b)}$ and $md\mid \dfrac b {\gcd(a,b)}$.

So $nd\cdot\gcd(a,b) = a$ and $md\cdot\gcd(a,b) = b$.

So $d\cdot\gcd(a,b)$ is a common divisor of $a$ and $b$.

Since $\gcd(a,b)$, is the greatest of those, we must have $d\not&gt;1$.