Solve for $(p,q)\in\mathbb{Z}$, $\frac{p}{\sqrt{3}-1}+\frac{1}{\sqrt{3}+1}=q+3\sqrt{3}$

Question

The question says "find integers $p$ and $q$ such that $\frac{p}{\sqrt{3}-1}+\frac{1}{\sqrt{3}+1}=q+3\sqrt{3}$.
I tried solving it but couldn't quite get the grasp of it.

It's solved. Thank you.

Answer 1

Hint: Assuming you mean $\frac{p}{\sqrt{3}-1}+\frac{1}{\sqrt{3}+1}=q+3\sqrt{3}$ you can multiply the terms in the left hand side by $\frac{\sqrt{3}+1}{\sqrt{3}+1}$ (first term) and $\frac{\sqrt{3}-1}{\sqrt{3}-1}$ (second) and then group terms in $\sqrt{3}$. You should get $p=5$ and $q=2$.