# number theory – Inequality first – why does it work?

The exercise is a following: Find all the first triplets $$(p, q, r)$$ so that $$pq + qr + pr> pqr$$. The solution: WLOG $$p leq q leq r$$. Yes $$pq + qr + pr> pqr$$then replacing $$p$$ gets $$3p ^ 2> p ^ 3$$ where we get $$p = 2$$ recital $$p$$ is a prime number. Then we get an inequality $$2q + 2r + qr> 2qr$$. Replacing $$q$$ we have $$q ^ 2 + 4q> 2q ^ 2$$ where we get $$q = 2$$ or $$q = 3$$. Now, substituting these values, we get that if $$q = 2$$, then all the values ​​for $$r$$ give solutions and if $$q = 3$$ then either $$r = 3$$ or $$r = 5$$. The answer is correct but why? Why can you replace $$p$$ and $$q$$ like that? Or is it just a coincidence?