Recently I came across a Wikipedia page concerning Modular Arithmetic. Some properties were stated there, but without any proof. I tried solving them. Most were solved but I got stuck up in two of them.
They are:-
1.If a ≡ b (mod n), then p(a) ≡ p(b) (mod n), for any polynomial p(x) with integer coefficients.
2.If c ≡ d (mod φ(n)), where φ is Euler's totient function, then a^c ≡ a^d (mod n)—provided that a is coprime with n.
I tried to prove them quite a few times, but failed.
Can someone please help me with these. The Wikipedia page is this.
Thanks in advance!
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Bill Dubuque
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Swastik Saha
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Welcome to Mathematics Stack Exchange. For 1., see this. For 2., did you mean $a^c\equiv a^d$? – J. W. Tanner Nov 03 '21 at 03:53
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$(1)$ is the Polynomial Comgruence Rule and $(2)$ is a special case of modular order reduction using Euler's (totient) Theorem. Follow the links for full proofs, and more. – Bill Dubuque Nov 03 '21 at 07:50
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By order, $$\begin{align*}a&\equiv_n b \\ \Rightarrow a^j&\equiv_n b^j\end{align*}\\ \Leftrightarrow ka^j\equiv_n kb^j,$$ and this for any $k,j\in\mathbb{Z}$ so you can basically make any poynomial that you want. For the second one I think that you are talking of $$c\equiv_{\varphi(n)}d\Rightarrow a^c\equiv_n a^d$$ if $(a,n)=1,$ this isn´t that hard to see, it´s only realizing what´s totient function structure.
