Hello everyone, I have been trying to teach myself number theory and I am stuck trying to prove a (I am sure) very easy to prove theorem related to that of Fermat's.(adsbygoogle = window.adsbygoogle || []).push({});

The theorem I am to prove states:

Let [tex]e[/tex] be the lowest number (natural) such that [tex] a^e \equiv 1 (\bmod \ p) [/tex] for [tex] p [/tex] prime such that [tex] p [/tex] does not divide [tex] a [/tex]. Prove that [tex] p-1 [/tex] is a multiple of [tex] e [/tex].

This theorem came after the proof of Fermat's Theorem, which I will also write for the sake of completeness.

Let [tex] p [/tex] be a prime and let [tex] a [/tex] be an integer such that [tex] p [/tex] does not divide [tex] a [/tex] it follows that:

[tex] a^{p-1} \equiv 1 (\bmod \ p) [/tex]

The book I am reading from (What is mathemathics? by Richard Courant and Herbert Robbins) suggests that I use the fact that [tex] a^{p-1} \equiv a^e \equiv 1 (\bmod \ p) [/tex] and that I divide [tex] p-1 [/tex] by [tex] e [/tex] to get [tex] p-1 = ke + r [/tex]

where r is the residue.

I've given it some thought but I kinda blow at math and I've done no useful advances, I was wondering if anyone could provide some more more insight.

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# Help with proof of theorem related to Fermat's.

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