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Let $a$ and $n$ be coprime. Then $$a^{\varphi(n)} \equiv 1 \mod n$$

Can you think of an easy, possibly visual, way to justify this theorem without delving into the group theory? In other words, if you were to explain the gist of this theorem to your 12-year-old sibling, how would you do it?

Aemilius
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    It happens that I have a 12-year-old daughter. But I think that it will be a waste of time to try to explain her why this is true. – José Carlos Santos Jun 04 '18 at 16:02
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    I think that anyone really needing, or being interested, in knowing "the gist of that theorem" would need to learn first the very small amount of group theory needed to understand it...Sometimes things cannot be dumbbed down that much. – DonAntonio Jun 04 '18 at 16:05
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    $x \mapsto ax$ is a bijection on coprimes $\bmod n$ - I would focus on that. This gets understood by "anyone" who just started studying this bit of ENT, and I don't see a way to put things simpler... – metamorphy Jun 04 '18 at 16:16
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    I’d do it using reduced residue classes (see https://math.stackexchange.com/questions/50542/proof-of-eulers-theorem-without-abstract-algebra) though it would require some further explanation on your part. Perhaps someone else would know how to adjust this to a 12 year olds level. – user328442 Jun 04 '18 at 16:18
  • In any case one has to introduce some elementary number theory. And this is already the way to abstract algebra. So why not start this way. If you are twelve, then you will also (usually) not be interested in arithmetic functions like $\phi(n)$. So I don't see too much sense in explaining this without any abstract algebra. – Dietrich Burde Jun 04 '18 at 16:55

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The easiest proof relies on $x \mapsto ax$ being a permutation of $U(n)$. That's the proof in Wikipedia.

This proof does not depend on Lagrange's theorem of group theory. It actually provides a simple proof of Lagrange's theorem for all finite abelian groups. Whether this proof can be explained to a child, I don't know.

lhf
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  • That $x\mapsto ax$ is a permutation could definitely be explained to a 12 year old, but would you mind expanding your answer or give a reference? – Cheerful Parsnip Jun 04 '18 at 16:21
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    See also https://math.stackexchange.com/questions/50542/proof-of-eulers-theorem-without-abstract-algebra. – lhf Jun 04 '18 at 16:23
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For 12 year olds rigorous deductive proofs are difficult I would go inductive. Start with a =2 and n as any prime. Most 12 year old should be able to figure this out .then generalize it to square free composite numbers as n. Then you can extend it to all odd numbers and proceed step by step . Will take some time but nothing difficult.