if $m$ is an integer and $\gcd(m,3)=1$，prove that $m^7\equiv m \pmod {63} $. I try using Fermat's Little Theorem,but it doesn't work. Please help me.
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Please write more informative titles when you ask a question (how to ask). This title is quite ambiguous and it could be about geometry. Remember that other users cannot see the tags before clicking on the question. – Toby Mak Oct 18 '20 at 06:03
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Sure,I will try to improve it. Thank you! – CHI NEW Oct 19 '20 at 04:42
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Hint $63=9*7$. Use Chinese Remainder Theorem.
N. S.
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Thank you, what you mentioned is the content of our next chapter, I will preview it and have a try. – CHI NEW Oct 18 '20 at 02:50
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With $\varphi(9)=6$, where $\varphi$ is the Euler Totient Function, and $(m,3)=1$: $$m^7\equiv m\mod 9$$ By FLT, we also have: $$m^7\equiv m\mod 7$$ Then using Chinese Remainder Theorem: $$m^7\equiv m\mod 63$$
Lynnx
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