# Tagged Questions

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### Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim

Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...
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### Is the reasoning/algebra for my proof correct? (musical tuning theory proof)

This isn't for a class, I was just wondering if I would be able to work out a proof for something like this myself for fun, and wanted to verify that my methods are correct. Basically, what I'm trying ...
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### Proof verification: Let $a$ be an irrational number and $r$ be a nonzero rational number. If $s$ is a rational number then $ar$ + $s$ is irrational

I have to prove the following: Prove: Let $a$ be an irrational number and $r$ be a nonzero rational number. If $s$ is a rational number then $ar$ + $s$ is irrational So, I decided to do a proof ...
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### Proof - Fundamental Theorem of Arithmetic using Euclid's Lemma

Let $n \in Z > 1$. Then the expression for $n$ as the product of $\ge 1$ primes is unique, up to the order in which they appear. From Proofwiki. Suppose $n$ has two prime factorizations: ...
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### Are these solutions to Linear Diophantine Equations too? Where'd they hail from?

(1) Can't the signs - I colored them in red - of x and y be switched? Aren't $x = x_0 - bn/d$ and $y = y_0 + an/d$ also solutions? They satisfy $ax + by = c$? (2) How can I remember these ...
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### My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
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### My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade.

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
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What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
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### Is this divisibility proof by induction correct/sufficient?

To show: 13 | $4^{2n+1}+3^{n+2}$ I used induction beginning successfully with n=0 (or n=1), then making the step to n+1: An x exists so that $13x = 4^{2n+3}+3^{n+3}$ $13x = 16*4^{2n+1}+3*3^{n+2}$ ...
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### A question on odd perfect numbers

Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. If $\sigma(M) = 2M$, then $M$ is said to be perfect. Currently, there are $48$ known examples of even perfect numbers -- on ...
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### How do $x^{p}+y^{p}=z^{p}$ and $x\equiv y \pmod{p}$ together imply $x\equiv -z \pmod{p}$?

I am reading Marcus' Number Fields and I have been a little stuck following his argument in page $4$ (where he is sketching an argument for Case 1 of Fermat's Last Theorem for primes $p$ for which ...
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### How to show that if $p \equiv 1,3 \pmod 8$ then there exists a $u,v \in \mathbb Z: u^2 + 2v^2 = p$

I'm trying to show this statement: $$p \equiv 1,3 \pmod 8, \; \; \exists \; u,v \in \mathbb Z : u^2 + 2 v^2 = p.$$ I believed I proved it the other direction using the ring $\mathbb Z{\sqrt{-2}}$. ...
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### Prove by induction that $3\mid (n^3 - n)$

I'm having an argument with my professor whether my exam was right or not. Before I sign a formal complain to get a review on my exam, I'd like to be sure it's correct. My answer: Proof by induction: ...
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### Prove directly, by contradiction, or contraposition? If the product of two integers is even, at least one of them must be even.

Hello I am having some trouble trying to know which way I should go as for this proof: ( we are suppose to use either direct, contradiction or contraposition) Prove or disprove: If the product of two ...
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### On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$

Note: This question was cross-posted from MO. Preamble: I apologize in advance if this particular MSE post would appear to be a bit of a polymath approach, I just had to put down all the details to ...
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### Show that if $a$ is an even integer and $b$ is an odd integer then $(a, b) = (a/2, b)$

Show that if $a$ is an even integer and $b$ is an odd integer then $(a, b) = (a/2, b)$ Hi everyone, I would like to know if my assumption is justified for answering the above question. Any ...
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### Show that if both a, b are even integers not both 0, then $(a, b) = 2 (a/2, b/2)$

Show that if both a, b are even integers not both 0, then $(a, b) = 2 (a/2, b/2)$. Hi there, I want to know if the following proof I have is strong enough, or if I'm making false assumptions :|. ...
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### Help with understanding this proof (I think it's Hensels Lifting?)

I am reading a proof that shows that if $a$ is a quadratic residue modulo $p^k$, where $p$ is a prime > $2$ and $k$ is a positive integer, then it is also a quadratic residue modulo $p^{k+1}$. Here ...
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### Proof of statement: If $a\mid b$ and $a\mid c$, then $a \mid b+c$

Statement: If $a$ divides both $b$ and $c$, then $a$ divides $b+c$ Proof: Assume that $a$ does not divide $b+c$. Then there is no integer $k$ such that $ak=b+c$. However, $a$ divides $b$, so $am=b$ ...
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### How to show that $(3k+2,5k+3)=1$ for all $k\in\mathbb{Z}$

I think I'm on the right track, but I can only figure out how to prove for a specific $k$ of my choosing... I don't know how to generalize it for all $k$: Assume $(3k+2,5k+3)=1$. Therefore, there ...
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### Prove that the Diophantine equation $ax+by+cz=e$ has a solution if and only if $(a,b,c)\mid e$.

I have an intuitive idea about how this is going to work, but I don't know if I'm writing it properly using proper math language and theorems. I am most uncomfortable with the second half of the proof ...
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### Can powers of primes be perfect numbers?

I need to prove the following, though I'm not 100% certain I understand the definition of a perfect number. Prove that no perfect number is a power of a prime. First of all, I'm assuming that ...
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### Number theory proof from AoPS

http://www.artofproblemsolving.com/Resources/articles.php?page=htw.readers In the above link, he gives a problem, namely Let $S(n)$ be the sum of the digits of $n$. Find ...
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### Proof Checking and input: Generators of $\mathbb{Z}_{pq}$

I'm self-studying abstract algebra (slowly but surely), and I have a question about my answer to the following prompt: Problem statement: Show that there are $(q-1)(p-1)$ generators of the group ...
Prove If n is an integer, then $n^2+n^3$ is an even number. I don't know if I'm just reiterating what I'm asked to prove or if my ideas are actually proving the statement. If $n^2+n^3$ is an even ...
The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \& \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$ Proof by Strong Induction : ...