How to figure out all possible pairs of numbers with a HCF?

Question

The product of two numbers is $13005$ and their HCF is $17$. Find all possible pairs of numbers.

I've done the first part of the question but I'm stuck on how to find all possible pairs of numbers. Is there any easy way to find all possible pairs of numbers with their HCF being $17$?

The number I'm trying to find all the combinations of is $765$

Please make it really simple for me as I struggle to understand things when its written in complex algebra form.

Thanks in advance

Answer 1

Hint: $ab=13005=45\times17^2$. Since $17$ is a common factor, we must have $a=17c$ and $b=17d$, where $cd=45$. Since it is the highest common factor, we need $c$ and $d$ to not have a common factor (if they have a common factor $h$ then $17h$ is a common factor of $a$ and $b$). How can you factorise $45=3^2\times 5$ so that the two factors are coprime?

Answer 2

$13005=3^2\times 5\times 17^2$.

\begin{array}{|c|c|c|c|} \hline \Large{17\times}& & \Huge{\times} & \Large{17\times} & \\ \hline & 3\times 3\times 5 & & & 1 & & Yes\\ \hline & 3\times 3 & & & 5 & & Yes\\ \hline & 3\times 5 & & & 3 & & No\\ \hline \end{array}