# Tagged Questions

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### Calculating $x^{103} \equiv 2 \pmod{143}$

I need to find x, given that: $0\leq x \leq 143$ and $x^{103}\equiv 2 \pmod{143}$. I tried to use Euler's theorem $p(143)=120$, but it didn't help.
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### Why is $U(12) = U_{4} (12) ~ U_3(12)?$

Why is $U(12) = U_{4} (2)~ U_3(12)$ Attempt: Any subgroup $U_k(n) = \{x \in U(n)~~|~~x \mod k=1 , k ~|~n \}$ Hence : if $U(12) =\{1,5,7,11\}$ then : $U_4(12) =\{1,5\}$ and $U_3(12)=\{1,7\}$ We see ...
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### How many subgroups there is for $\mathbb{Z}_{23}\times \mathbb{Z}_{31}$?

How many subgroups there is for $\mathbb{Z}_{23}\times \mathbb{Z}_{31}$? I try to understand how to solve it but I don't finding a way... I'll be glad if you help me with this... Thank you!
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### Find period of power sequence $a^k \mod m$, with $a, m$ not coprime

Let $a, m$ be positive integers and $m > 1$. I'm interested in the sequence $(a^k)_{k \in \mathbb{N_0}} \mod m$. Since there are only $m$ different values that can occur in the sequence and since ...
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### Finding subgroups via modulo arithmetic [closed]

Consider the following numbers : $1, 3, 7, 9, 11, 13, 17, 19$. In terms of modulo $20$ how many subgroups are there? For example, $3(7) = 21 = 1$ mod $20$. In particular I want to find the ...
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### A nice group isomorphism

Show that $$k(\mathbb{Z}/n\mathbb{Z})\cong (\gcd(n,k)\mathbb{Z})/n\mathbb{Z}.$$ I want to see as many as possible proofs of this nice fact.
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### How do I prove that $a\in (\mathbb{Z}_{p}^{*})^2 \Leftrightarrow a^{\frac{p-1}{2}}\equiv 1 \pmod p$

$p$ is prime number $>2$ and $a$ is a square. $\mathbb{Z}_{p}^{*}$ is a cyclic group. I need to show that $$a\in (\mathbb{Z}_{p}^{*})^2 \iff a^{\frac{p-1}{2}}\equiv 1 \pmod p$$ Any ideas ...
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### Fermat's little theorem for $n=3$
for $N > 0$, I'm trying to show Fermat's little theorem, for $3$ using the orbit stabilizer theorem: $N^3 - N$ an element of $3\mathbb{Z}\ (3 \mod \mathbb{Z})$ Pf/ we can break it down into ...