# Questions tagged [gcd-and-lcm]

Use for questions related to gcd (greatest common divisor), lcm (least common multiple), and related concepts from integer and ring theory.

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### What I am doing wrong when trying to solve: if $\gcd(a,b) = 1$, then $\gcd(2a+b,a+2b) \in \{1,3\}$?

First of all, I want to acknowledge that I had many of my questions closed because of duplicates. I am aware that this problem has been posted many times(here, here,here). I am not trying to get a ...
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### Space obtained by attaching two $2$-cells to $S^1$

I am working on the following problem: Let $X$ be the CW complex obtained from $S^1$ by attaching two $2$-cells: one by a map of degree $2$, and one by a map of degree $3$. (a) Compute the homology ...
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### Proofs regarding gcd

I have two questions about proofs regarding gcd. We have $x^2+y^2+z^2=m^2$ with $x,y,z,m \in \mathbb{N}$. 1.) I want to show that, $gcd(x,y,m,z)=1$ is equivalent to $g(x,y,z)=1$ My attempt: The back ...
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### Maximum number of colours we can use to colour all the integers so that for any integers i,j, i and j have the same colour if |i-j|=38 or |i-j|=27?

I tried solving this by plugging in a few smaller values for the differences, and it turns out that whenever the differences are coprime numbers, 1 colour is required. However, I couldn't construct ...
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### Is there a name for $\text{LCM}(a, n)/a$?

Unless I am mistaken, for any $a, n \in \mathbb N$ the following are equivalent: $\text{LCM}(a, n)/a$ The minimal $\ell \in \mathbb N$ such that $\underbrace{a + a + \cdots + a}_{\ell \text{ times}}$...
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### Calculate $\gcd(2a+4b, 2a +8b)$, if $a\equiv b \pmod{\! 5},\ 6a+11b = 5$

Given: $a$ is even $6a+11b=5$ $a-b=0\pmod 5$ Q: Calculate $\gcd(2a+4b,2a+8b)$ My try: We know there is some $i$ such that $a=2i$, plus from 3 we know there is some $j$ such that: $a-b=5j$ which ...
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### Does $\sigma(m^2)/p^k$ divide $m^2 - p^k$, if $p^k m^2$ is an odd perfect number with special prime $p$?

The following query is an offshoot of this post 1 and this post 2. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. The topic of odd perfect numbers likely ...
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### Finding the $\gcd$ of two polynomials over $\Bbb Z[x]$

I understand that there are other posts on the forum about the same topic, but, after reading them, I didn't understand exactly what to do in this situation. What I've done so far. In this same ...
A pair of integers $(x,y)$ can be reduced to a pair of coprime integers via $(x_1,y_1) = \left(\frac{x}{d},\frac{y}{d}\right)$ where $d = gcd(x,y).$ Suppose we have a triplet of integers $(x,y,z)$ and ...
I refer to a previous post of mine in which it is defined the recurrence relation (now OEIS sequences A349576 and A349982) $$x_{n+1} = \frac{x_{n} + x_{n-1}}{(x_{n},x_{n-1})} + c\;\;\;\;\;\;\;(1)$$ ...