the answer is (1,2) and (2,7) but how do you prove they are the only pairs

pl ping me if you can help me with this @user21820

This is an instance of the canonicalization technique, which you should learn to employ whenever possible; canonicalization just means to restrict your attention to some kind of canonical form that you can reduce every other case to. Of course, you have to figure out what is suitable to be the canonical form, but the point is to have the correct mindset in the first place.

That is, you should already think about the useful canonical forms for anything you come across even before you face problems involving them. For instance, you came across fractions of polynomials. According to this mindset, you should have already thought about what canonical forms are convenient. One of them is what I described above, which is useful not just here but also when you want to integrate via partial fractions.

Sadly, this kind of mindset is not something that most teachers are aware of, much less teach students to learn. So I'm not expecting you to have learnt this before, and I hope you can see from my explanation why it is useful to train yourself to have such a mindset.

Wow !! I never would have thought this, could you give a reference for where I can read more about this cancolization technique @user21820

@Buraian I figured it out myself, so I don't have a reference for you. But there is a tendency to learn such things if you spend some time thinking about how to prove identities. One way is to reduce to as 'simple' form as possible, and if there is a unique 'simplest' form by some measure of 'simple' then that can be defined as a canonical form. Of course, canonical forms need not look simple; it's up to you to define it. But generally you want to reduce to some minimal representation.

For example, if to prove a high-school trigonometric identity you can firstly replace all "tan" by "sin/cos" and "cot" likewise, and then convert to polynomial in terms of sin,cos and then use the angle-sum identities and so on, all with the goal of 'meeting in the middle'. The very action of eliminating "tan" and "cot" is part of the canonicalization mindset.

I had a vague notion of what you're saying before you said it itself but it's like you have put the thoughts in my head into precise words. Very good, I think I'll try to introduce this idea in future answers I write so more people will know about it