Questions tagged [gcd-and-lcm]
For questions related to gcd (greatest common divisor, also known as the hcf, the highest common factor), lcm (least common multiple), and related concepts from integer and ring theory.
2,709
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$x=\frac{(2^a)(z)-(3^b)(w)}{(2^a)-(3^b)}$ does any natural odd number value of x >1 staisfy this equation [closed]
here z and w are integer and a and b and x are natural number
i forget to mention
$w = x-(2^a)$
$z = x-(3^b)$
which make gcd(z,w)=1
i know a set of solution for x = 1 which is
w = -3
z = -8
a = 2
b = ...
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Find the greatest common divisor of the numbers: $2^{2^{2019}}-1, 2^{2^{2021}}-4$. [duplicate]
Problem
Find the greatest common divisor of the numbers: $2^{2^{2019}}-1, 2^{2^{2021}}-4$.
My Idea
To be honest, I didn't know how to start so I let $d$ be the greatest common divisor of those numbers
...
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How to find the largest set of coprimes between two numbers M and N?
First, consider this simpler problem:
Given a number $N$, find the maximum set of co-primes $A$ (i.e. $\forall a, b \in A\implies \text{GCD}(a, b) == 1$) such that each number $a\in A$ is $a < N$.
...
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Does the following GCD divisibility constraint imply that $\sigma(m^2)/p^k \mid m$, if $p^k m^2$ is an odd perfect number with special prime $p$?
The topic of odd perfect numbers likely needs no introduction.
In what follows, denote the classical sum of divisors of the positive integer $x$ by $$\sigma(x)=\sigma_1(x).$$
Let $p^k m^2$ be an odd ...
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Interesting NT Question With AP and GCD.
Find the number of prime triplets $(p, q,r)$ such that $p(p + 1), q(q + 1),r(r + 1)$ form a strictly increasing arithmetic progression, where GCD $(r − p, 2p + 1)=1$.
What I tried:
$r(r+1)-p(p+1)=2d$ ...
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What $n$ would make $ \gcd _{k\ge1 }\left\{\dbinom{m+(k-1)\cdot n}{k}\right\}=1 \ \forall m \in \mathbb{N}$
This is a generalisation of this question.
Construct the sequence $a_n$
$$ a_n =
\begin{cases}
0, & \gcd _{k\ge1 }\left\{\dbinom{m+(k-1)\cdot n}{k}\right\}=1 \ \forall m \in \mathbb{N} \\
\min{m ...
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Conventional notation for gcd and principal ideal in the context of Bezout domain [duplicate]
Background
Definition 1: Let $R$ be a commutative ring with identity, $c\in R$ and let $I$ be the set of all multiples of $c$ in $R$, that is, $I=\{rc\mid r\in R\}$. Set $I$ is an ideal and is ...
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Slope and GCD points
given a slope in the form $y=ax+b$
My question is to know if there are (and if yes how to find them) points $(x,y)$ where :
$gcd(x,y)>1$ (to be more precise, the $gcd$ is made between the $x$ and ...
4
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On Prime, Lcm and Divisibility
Let $d_n= \text{lcm} \{1,2,3..., n \}$ then prove that for any prime $p$ $$ \left(\frac{1}{p^5-1}\right)\left(\frac{d_{p}}{p}\right)^5 \text{is not an integer}$$
Clearly $$\frac{d_p}{p}\in\mathbb{Z},...
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2
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Expected number of factors of $LCM(1,…,n)$ (particularly, potentially, when $n=8t$)
I’m trying to prove something regarding $8t$-powersmooth numbers (a $k$-powersmooth number $n$ is one for which all prime powers $p^m$ such that $p^m|n$ are such that $p^m\le k$). Essentially, I have ...
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$\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$?
I am exploring whether the following assertion holds true in integral domains: $\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$. Let us make this formal below.
Consider two elements $a$ and $b$ in an ...
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$f((x,y))=(x*gcd(x,y),y) $is injective and surjective?
The function $f:\mathbb{N^+}\times\mathbb{N^+} \rightarrow \mathbb{N^+}\times\mathbb{N^+} $ definide as $f((x,y))=(x*gcd(x,y),y) $is injective and surjective?
Let $(a,b)\in\mathbb{N^+}\times\mathbb{N^+...
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Question on LCM and divisibility
Let $d_n= \text{lcm} \{1,2,3..., n \}$ then for which natural numbers $n$ is $$\frac{24 d_n^5}{(n+1)^5} \text{not an integer?}$$
We know that $$ \text{lcm}\{1,2,3,...,n\}.\text{gcd}\{1,2,3,...,n\}=n!$...
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Velleman 6.4.17 - Stuck with proof involving GCD and divisibility
Level and context: I'm a first-year undergraduate self-studying proof writing from Velleman's How to Prove It (second edition).
Exercise 17 (a). Suppose $a$, $b$, and $p$ are positive integers and $p$ ...
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In $\Bbb Z[i]$ find $\gcd(2-7i,2+11i).$ Also find $x,y\in \Bbb Z[i]$ such that $(2-7i)x+(2+11i)y=\gcd(2-7i,2+11i).$ [duplicate]
In $\Bbb Z[i]$ find $\gcd(2-7i,2+11i).$ Also find $x,y\in \Bbb Z[i]$ such that $(2-7i)x+(2+11i)y=\gcd(2-7i,2+11i).$
I tried solving the problem as follows:
Let $d=\gcd(2-7i,2+11i).$ So, $$d|2-7i,d|2+...