I'm reading a book on number theory (*Theory of Numbers, Niven*), and yesterday I've stumbled upon a proof of the above lemma (*Lemma 2.13; page 54-55*). I've managed to wrap my mind around the proof from the book, but it is really unintuitive and hard to follow(at least for me).   

My question to you guys is do you know of any simpler and more intuitive proof of that particular lemma?   

edit: here's the link for the proof from the book   

![enter image description here](https://i.stack.imgur.com/slEgL.jpg)