Show that two numbers divided by their GCD are coprime

Let $a, b \in \mathbb{Z} \setminus \{0\}$ and $d = gcd(a, b)$. Show that $gcd(\frac{a}{d}, \frac{b}{d}) = 1$.

I tried proving this by contradiction and showing that otherwise $d$ isn't the gcd of $a$ and $b$, but it didn't work. Could someone please give me a hint on what the proof should look like?

Answer

If $d'>1$ divides both $\frac{a}{d}$ and $\frac{b}{d}$ then $dd'> d$ divides $a$ and $b$, contradicting the fact that $d$ is the greatest common divisor of $a$ and $b$.