I would like to prove it using Euler's Theorem, which states that if $\gcd(a,n)=1$,then $a^{\phi(n)}=1 \pmod{n}$. Then, $480$ should be factored, but I am unsure how to proceed with the proof. Thanks in advance.
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Did you at least calculate $\phi(480)$? – Ben Grossmann Nov 10 '20 at 01:57
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Yes! $\phi(480)=128$, but I do not understand how that helps. – qt_314 Nov 10 '20 at 01:59
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@qt_314, s use https://mathworld.wolfram.com/CarmichaelFunction.html – lab bhattacharjee Nov 10 '20 at 02:08
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Same proofs as in the dupe work here. – Bill Dubuque Nov 10 '20 at 02:12
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@BillDubuque Thank you so much! I apologize for posting a duplicate question. – qt_314 Nov 10 '20 at 02:14
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It's best for site health to delete dupes of FAQs (we have many tens if not hundreds of posts on this topic - little new can be said). The dupes make it harder to locate the best answers when searching (which you should always do before asking a question). – Bill Dubuque Nov 10 '20 at 02:16
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@BillDubuque I will certainly delete it then. First, if you don't mind me asking, what did you search to find the duplicate? – qt_314 Nov 10 '20 at 02:18
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I used keywords like Euler. phi, carmichael but you could also search on equations using https://approach0.xyz/search/ – Bill Dubuque Nov 10 '20 at 02:20
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@BillDubuque Thanks again. Just tried to delete the post, but it said I cannot since others have already answered it. Is there something else I should do? Sorry, I am new to this site. – qt_314 Nov 10 '20 at 02:22
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Hint: $480 = 2^5 \cdot 3 \cdot 5$. By the Chinese remainder theorem, we see that $a^k \equiv 1 \pmod {480}$ if and only if we have $$ \begin{cases} a^k \equiv 1 \pmod{2^5},\\ a^k \equiv 1 \pmod 3,\\ a^k \equiv 1 \pmod 5. \end{cases} $$
Ben Grossmann
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@qt_314 The approach I suggest to this problem does not involve Euler's theorem at all, if that's what you mean. When I made my earlier comment, I had not yet thought of this approach. – Ben Grossmann Nov 10 '20 at 02:03
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@qt_314 See also here for motivation on the relationship between little Fermat and Euler. – Bill Dubuque Nov 10 '20 at 02:25