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I just began learning elementary number theory and I am still very new in terms of symbols and existing proofs/theories. I would like to learn to prove that Euler's Totient Function is multiplicative. I have found a resource that proves that Euler's Totient Function is multiplicative, though there is an extra paragraph that I don't understand, nor see why it would is required to fulfill the proof. I believe that the Lemma in combination with a part of the theorem that follows, is enough to prove it is multiplicative. The extra paragraph (screenshot 2) contains the whole theorem with the lemma that the book claims fulfills the proof. Is it sufficient to just use the information from screenshot 1?

Lemma with part of the theorem (is this sufficient for proving it is multiplicative?)

Lemma with full theorem (or do I need the extra paragraph and information at the bottom?)

If you do find that the lemma with the full theorem is imperative, then I would appreciate an explanation. I am still a beginner.

Bill Dubuque
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tiktoker123
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  • If you are a beginner, I would recommend you take a step back, and focus on understanding how to apply the theorem. Take a look at the worked solutions on Brilliant and see if you can answer the questions. The proofs are not as important at this stage as they are just there for rigour: you might find yourself lost in the details if you try to analyse it too closely. – Toby Mak Oct 16 '20 at 12:21
  • @TobyMak is there a way I can do that without learning about rings? I have looked at brilliant and all i want is to understand this. – tiktoker123 Oct 16 '20 at 12:25
  • Just ignore the part about rings. The rest is quite useful. – Toby Mak Oct 16 '20 at 12:27
  • I do understand that the Brilliant article does not include a proof about why we can assume the totient function is multiplicative (this is what your textbook is for), but hopefully you can see how this approach can be used to solve problems. – Toby Mak Oct 16 '20 at 12:30
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    I vehemently disagree with using the Brilliant wiki as a resource. When I visited some pages a while ago, it was rife with small errors that added up to the point that I ceased to visit the website. – Favst Oct 16 '20 at 14:22
  • If by rings you mean modular arithmetic or extensions, I am aware of no finer resource than Niven's final edition of Introduction to the Theory of Numbers. – Favst Oct 16 '20 at 14:24
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    Please don't radically change your question (esp. after an answer has been posted). I have reverted it to the original version. If you have another question then post it anew. – Bill Dubuque Oct 17 '20 at 09:22

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Sometimes it is useful to read another text. There are very elementary proofs available at this site to show that $\phi(mn)=\phi(m)\phi(n)$ for coprime $m$ and $n$. The answers at the post below also include a solution using rings, but the other ones (and I count 13 of them) are only using very elementary steps. So you have the choice which one you like most.

What's the proof that the Euler totient function is multiplicative?

Dietrich Burde
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  • The answer refers to the original question (before the edit), why Euler's totient function is multiplicative. For rings in number theory and much more, I can recommend the book by Ireland and Rosen. It will be perfect for you. – Dietrich Burde Oct 16 '20 at 15:04