Can anyone please help me with this homework question on automata from Peter Linz? Use induction on $n$ to show that $|u^n| = n|u|$ for all strings $u$ and all $n$.

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Prove that that $U(n)$, which is the set of all numbers relatively prime to $n$ that are greater than or equal to one or less than or equal to $n-1$ is an Abelian ...

Mar 27, 2019 · I know the proof using binomial expansion and then by monotone convergence theorem. But i want to collect some other proofs without using the binomial expansion. *if you …

Right! I like that: $ (n-1)^2=n^2-2n+1=n (n-2)+1 \equiv 1 (\bmod {n})$. I was skeptical of the line "However, we know (I forgot the theorem's name) that the number of elements of order 2 is …

Thanks for the link @muzzlator. I've just had a look at it and it's very interesting (and seems a lot simpler), however it uses methods a little different to those that I have been using for the …

(If you know about ring theory.) Since $\mathbb Z_n$ is an abelian group, we can consider its endomorphism ring (where addition is component-wise and multiplication is given by …

@Lhf The question is certainly not a duplicate of the linked question, since the author is asking additionally a more general question, namely "What are those number theoretic situations?" …

I was playing with my calculator when I tried $1.5!$. It came out to be $1.32934038817$. Now my question is that isn't factorial for natural numbers only? Like $2!$ is $2\\times1$, but how do …

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