Question

Suppose $(n,m)=1$. Prove any automorphism of $\mathbb{Z}_n \times \mathbb{Z}_m$ is of the form $\varphi(x, y) = (ax, by)$.

score 60 · Answer 1 · 2026-03-29 12:11:18

60

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Suppose $(n,m)=1$. Prove any automorphism of $\mathbb{Z}_n \times \mathbb{Z}_m$ is of the form $\varphi(x, y) = (ax, by)$.

Published on 29 Mar 2026 - 12:11

score 56 · Answer 2 · 2026-04-13 17:39:36

56

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Congruence: How to prove a set assumes all values?

Published on 13 Apr 2026 - 17:39

score 65 · Answer 3 · 2026-04-16 14:12:13

65

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Order of elements and center of a subgroup of $GL_2(\mathbb{F}_7)$

Published on 16 Apr 2026 - 14:12

score 5.4k · Answer 4 · 2026-04-15 18:20:27

5.4k

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Let $G=\{-1,1,i,-i\}$ be the fourth roots of unity. Prove $G$ is a group under multiplication and $G$ is isomorphic to $\Bbb{Z}_4$..

Published on 15 Apr 2026 - 18:20

score 1.2k · Answer 5 · 2026-04-15 03:48:37

1.2k

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Multiplicative Inverse for Elements in an Integer Ring

Published on 15 Apr 2026 - 3:48

score 98 · Answer 6 · 2026-04-17 07:13:05

98

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How many roots has the equation $x^3+x^2+x+1=0$ modulo $340$?

Published on 17 Apr 2026 - 7:13

score 101 · Answer 7 · 2026-04-17 07:26:26

101

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$ 1 + 2^{p-2} + 3^{p-2} + \cdots + (p-1)^{p-2} \equiv 0\pmod p $ with $ p $ odd prime

Published on 17 Apr 2026 - 7:26

score 51 · Answer 8 · 2026-04-16 01:14:22

51

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Prove for every $k\geq1$ there exist $n$ that satisfies $2^n = -2^k$ (mod $10^k$).

Published on 16 Apr 2026 - 1:14

score 44 · Answer 9 · 2026-03-26 18:49:45

44

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Proof relating to equivalence classes

Published on 26 Mar 2026 - 18:49

score 32 · Answer 10 · 2026-04-11 03:02:12

32

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Can this congruence rule be generalized?

Published on 11 Apr 2026 - 3:02

score 31 · Answer 11 · 2026-04-12 15:09:55

31

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is mod statement correct

Published on 12 Apr 2026 - 15:09

score 1.5k · Answer 12 · 2026-04-15 14:06:38

1.5k

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Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.

Published on 15 Apr 2026 - 14:06

score 432 · Answer 13 · 2026-04-16 00:56:16

432

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Reduced residue classes the same as multiplicative group of integers?

Published on 16 Apr 2026 - 0:56

score 27 · Answer 14 · 2026-04-10 19:54:40

27

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Do modular rules apply for repeated exponents

Published on 10 Apr 2026 - 19:54

score 144 · Answer 15 · 2026-04-12 21:48:40

144

Views

How can you prove the existence of an integer for ab congruent to 1 (mod p).

Published on 12 Apr 2026 - 21:48

List Question

Suppose $(n,m)=1$. Prove any automorphism of $\mathbb{Z}_n \times \mathbb{Z}_m$ is of the form $\varphi(x, y) = (ax, by)$.

Congruence: How to prove a set assumes all values?

Order of elements and center of a subgroup of $GL_2(\mathbb{F}_7)$

Let $G=\{-1,1,i,-i\}$ be the fourth roots of unity. Prove $G$ is a group under multiplication and $G$ is isomorphic to $\Bbb{Z}_4$..

Multiplicative Inverse for Elements in an Integer Ring

How many roots has the equation $x^3+x^2+x+1=0$ modulo $340$?

$ 1 + 2^{p-2} + 3^{p-2} + \cdots + (p-1)^{p-2} \equiv 0\pmod p $ with $ p $ odd prime

Prove for every $k\geq1$ there exist $n$ that satisfies $2^n = -2^k$ (mod $10^k$).

Proof relating to equivalence classes

Can this congruence rule be generalized?

is mod statement correct

Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.

Reduced residue classes the same as multiplicative group of integers?

Do modular rules apply for repeated exponents

How can you prove the existence of an integer for ab congruent to 1 (mod p).

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