Number Theory - How to Prove n^7 is Congruent to n Mod 63

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SUMMARY

The discussion focuses on proving that for any integer \( n \) not divisible by 3, the congruence \( n^7 \equiv n \mod 63 \) holds true. The proof is established by separately demonstrating \( n^7 \equiv n \mod 7 \) using Euler's theorem. The challenge lies in proving \( n^7 \equiv n \mod 9 \), which requires understanding Euler's totient function \( \varphi(9) \) and its implications according to Euler's theorem.

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number theorem -- Euler theorem

Homework Statement



let be an integer that not divisible by 3. Prove that n^7\equivn mod 63

Homework Equations



none

The Attempt at a Solution


it is suffice to prove that n^7\equivn mod 7,n^7\equivn mod 9, i get
n^7\equivn mod 7 by Euler theorem , how to prove n^7\equivn mod 9
 
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Remember that Euler's totient function, \varphi (n) is equal to the number of positive integers less than or equal to n that are coprime to n. What is \varphi (9) and what does that imply by Euler's Theorem?
 

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