This tag is for questions about divisibility, that is, determining when one thing is a multiple of another thing.

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160 views

Does $A193201$ count the partitions of $n$ of arbitrary dimension?

By my count, these sequences match for $n=0\ldots6$, where partitions that are the same after relabeling dimensions are considered equivalent (i.e., the dimensions are unordered). For example, for ...
6
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171 views

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 ...
6
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0answers
187 views

What's the most efficient algorithm for Divisibility?

What is the most efficient (in time complexity) algorithm known nowadays for the Divisibity Decision Problem: given two integers, say $a$ and $b$, does $a$ divide $b$? Let it be clear that what I ask ...
5
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0answers
74 views

Product of consecutive Fibonacci numbers divisibility

Prove that the product of any $k$ consecutive Fibonacci numbers is divisible by the product of the first $k$ Fibonacci numbers. We can try to show that for every prime $p$, the power of $p$ appearing ...
5
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0answers
337 views

Natural extension of the divisor function $\sigma$ to the complex integers $\mathbb{Z}[i]$

The divisor function $\sigma$ of a positive integer $n$ is defined as the sum of the positive divisors of $n$. It turns out that this definition is equivalent to ...
4
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46 views

Extention of Euclid's GCD Algorithm. (The Art of Computer Programming, Volume 1, Edition 3, Section 1.2.1, Exercise 12)

Euclid's GCD algorithm which is used to find GCD of two input numbers, say, $c$ and $d$, needs the inputs to be positive integers. Exercise 12 provides an extension to this algorithm and allows $c$ ...
3
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0answers
39 views

Trying to prove a congruence for Stirling numbers of the second kind

I am struggling with a demonstration for this: When $n$ and $m$ are 2 natural integers such that $n-m$ is odd, then the following congruence holds for Stirling number of the second kind ${n \brace ...
3
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53 views

Generating all lesser numbers of two coprime numbers

Let's say I have two coprime positive integers, $a$ and $b$. How would you go about proving that it is possible to make all integers between 1 and $max(a,b)$ by subtracting them from each other? For ...
3
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368 views

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 :|. ...
3
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0answers
79 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
2
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30 views

divisibility and k-power sum

Let $a_{1},\dots,a_{n},\,n>2$ distinct natural numbers. Prove that if $p_{1},\dots,p_{r}$ are prime numbers and they divide $a_{1}+\dots+a_{n}$ then exists an integer $k>1$ and a prime $p\neq ...
2
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0answers
22 views

How do I know the cyclic subgroups of G generated by a are equal?

I had a question of understanding that I was hoping you guys could help me with. Suppose there exists a group in the integers. For the sake of my understanding, let's say the group is, G = $Z_{60}$. ...
2
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0answers
37 views

Is there a cycle for modulo operation on a floor?

I have a floor $${\left\lfloor\frac{n}{i}\right\rfloor},$$ where i varies from 1 to n (n can be upto $10^{10}$), and there's another number given 'm' (m upto $10^{5}$). Does ...
2
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0answers
79 views

How to find the approximate basic frequency or GCD of a list of numbers?

I could't actually summarize the question in the title, so I'll explain my situation. I want to tell the integer numbers which act as the best approximate basic frequencies of a list of real numbers: ...
2
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28 views

Prove the congurence

I am looking for a proof of Gauss's generalization of Wilson's Theorem. Let $S$ be the set of all the integers which are less than and mutually prime to $n (>4)$ (not of the form $p^\alpha$, ...
2
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50 views

Proof relating to Euclidian Algorithm

The question is as follows: (1): Let m and n be positive integers with n < m and let r be the remainder when m is divided by n. Prove that $$r < \frac m2$$ (2): The Euclidean Algorithm for ...
2
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66 views

$27^{2004} + 22^{2004} - 4^{2004} - 1$ is divisible by (options)

(A) $299$ (B) $296$ (C) $298$ (D) $297$ This kind of sums are too problematic. Please provide a method which could give the correct answer in about a minute. :)
2
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72 views

Find Gcd summation fast?

Find the value of the summation: $$ val=\left( \sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^c....\sum_{x=1}^p GCD(i,j,k,..x) \right)$$ Contraints $2\leq$number of summation terms$\leq 500$, $1\leq ...
2
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82 views

Fast check If the remainder is 1

Is there any fast method to 'say' that $R = (A \mod B)$ is $1$ or $R > 1$ or $R \neq1$ or $R > k>1$ ( where $k$ is a small integer on $32$ bits) without to actually calculate the real value ...
2
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0answers
101 views

When is a number like “ddd…ddd”+1 (where d is a digit) a perfect square or a prime?

Inspired by Is the number $333, 333, 333, 333, 333, 333, 333, 333, 334$ a perfect square?, I wonder when numbers like these are perfect squares. Certainly, all numbers of the form $000...0001$ are ...
2
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57 views

Prove that if $d_1=\gcd(a,b), d_2=\gcd(b,c), d_3=\gcd(c,a), D=\gcd(a,b,c)$, and $L=\operatorname{lcm}(a,b,c)$, then $L= \frac{abcD}{d_1 d_2 d_3}$

I tried to define: $a=d_1x_1$, $b=d_1y_1$; $b=d_2x_2$, $c=d_2y_2$; $c=d_3x_3$, $c=d_3y_3$. then $\operatorname{L.H.S} =d_1d_2d_3x_1x_2x_3=d_1d_2d_3y_1y_2y_3$ $\implies$ $x_1x_2x_3=y_1y_2y_3$; ...
2
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113 views

An upper bound on the least common multiple of the first $2n+1$ integers

Let $p$ be a prime number and let $a, n \in \mathbb{N}$. Then $$ p^a \mid \operatorname{lcm}(1, 2, \dots, 2n+1) \implies p^a \leq 2n + 1 \implies a \leq \dfrac{\ln(2n+1)}{\ln p}$$ and ...
2
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104 views

What is in common between decimals and reminder

I am trying to understand this solution for Project Euler' 26 problem. Could you please explain what do the decimals and remainders have in common? In context of this solution. Thank you in advance.
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163 views

Count arrays with GCD as D

Given N ,I need to count the number of array of integers which satisfy the following conditions : ...
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25 views

On counting number pairs having a specific greatest common divisor.

I wanted to count natural numbers $k$ not exceeding the fixed $n \in \mathbb{N}$ and having a greatest common divisor $\gcd(n,k) = d$ naturally for some $d \mid n$. In more mathematical terms: $$ ...
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0answers
40 views

If p is a prime what values of $ a\leq p^{n}$ have $\text{gcd}(a, p^{n}) >1$?

Hi guys need your help. Sorry but I don't understand how to use latex. So really sorry for the writing. The question is if p is prime what values of $ a\leq p^{n}$ have $\text{gcd}(a, p^{n}) >1$? ...
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22 views

Using divisibility and greatest common divisor for a proof

If u|t and v|t and gcd(u,v)=1, then prove that (uv)|t I started by analyzing the definition of divisibility and I got that (uv)|t^2, but this doesn't help me. Any advice would be appreciated. Thank ...
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21 views

GCD-Domain and proprieties

Let $A$ be a commutative GCD-domain (not necessary UFD or Bezout) and $a,b,c$ elements of $A$ such that $\gcd(a,b) = \gcd(b,c) = \gcd(a,c) = 1$. Is it true that $\gcd(ab,c) = 1$ ?
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28 views

Prove that $10 | (n^a - n^b)$.

$n$ is a positive integer. Prove that there exists positive integers $a$ and $b$, $(a > b)$ such that $10 | (n^a - n^b)$. I have tried to prove this by induction on $n$, but I get stuck at the ...
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0answers
74 views

Any general hints on how to prove that two functions$\ f(n)$ and$\ g(m_1,m_2,…,m_{28})$ never have a common natural divisor?

All the variables are natural numbers. I'm not asking for a proof, since while we simply have$\ f(n)=n^3-n+1$,$\ g$ is a very long sum of cube roots (which contain square roots as well). I'm after ...
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21 views

Sequence terms being divisible

Here's a question I would like hints for: The sequence ${x_n}$ is defined by $x_{n+2}=6x_{n+1}-9x_{n}$ for $n \geq 0$ where $x_0=3$ and $x_1=18$. What is the smallest $k$ such that $x_k$ is ...
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40 views

A Question Related to Zsigmondy's Theorem

I am wondering if there is a way to prove the following statement, which bears some resemblance to Zsigmondy's Theorem. I am not sure if the statement is true, but it seems as though it should be. ...
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53 views

$U(n) =$ the smallest strictly positive integer $k$ such that $2^k-1$ is divisible by the first $n$ primes. What are the first values of $U(n)$?

$U(n) =$ the smallest strictly positive integer $k$ such that $2^k-1$ is divisible by the first $n$ prime numbers (except for the first prime number: $2$). What are the first values of $U(n)$ up to ...
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33 views

Find the Conditions

Let $a, b, c, d, r, s \in \mathbb{N}$. Find the necessary and sufficient conditions under which $r \mid (a-b)$ and $s \mid (c-d)$ $\implies$ $\operatorname{lcm}$ $(r,s)\mid(ac-bd)$. A little ...
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63 views

Showing DO NOT exist GCD of $6$ and $2+2 \sqrt{-5}$ in $\Bbb Z[\sqrt{-5}]$.

Showing DO NOT exist gcd of $6$ and $2+2 \sqrt{-5}$ in $\Bbb Z[\sqrt{-5}]$. I tried it. Suppose $d$ is GCD of $6$ and $2+2 \sqrt(-5)$. then there exist $x,y \in \Bbb ...
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14 views

Efficient way needed

Given N, M find the : GCD(1, 1) * GCD(1, 2) * … * GCD(1, M) * GCD(2, 1) * GCD(2, 2) * … * GCD(2, M) * … * GCD(N, 1) * GCD(N, 2) * … * GCD(N, M) modulo 10^9+7 Constraints: 1 <= N, M <= 2 * ...
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0answers
27 views

Using $ \gcd(a,b) = \gcd(b,r) $ if $ a \equiv r \pmod b$ for GCD?

It should be true that $\gcd(a,b) = \gcd(b,r) $ if $ a \equiv r \pmod b$. But: How can I use this equality to compute the GCD of $a$ and $b$? It seems as if $r$ is of the form $r = k\cdot b + s$ ...
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50 views

GCD of this polynomial

So here is the exact question that i am having trouble on: "Extend the Euclidean algorithm to polynomials and find the greatest common divisor of: $3x^5-10x^4-4x^3-14x^2-7x-4$ and ...
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32 views

find all the divisors of $6$ and $4+2\sqrt{5}$,then find $\gcd(6,4+2\sqrt{5})$

By inspection we see that the divisors of $6$ are $1,2,3,6$ For $4+2\sqrt{5}$ we have $4+2\sqrt{5}=2(2+\sqrt{5})$ showing that $\gcd(6,4+2\sqrt{5})=2$ Is this method correct; if not, how can I do ...
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39 views

Congruence equations

Given positive integer $Z, N$ and a set of positive integer $S$. Find smallest $k \in \mathbb{Z^+}$ such that $$a*k +1 \equiv Z \pmod N \ a\text{ is a positive integer that we don't know, and}\\ i*k ...
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90 views

How do you justify the PigeonHole principle?

I am working on the problem below and just have two questions pertaining to my answers. 1) Am I clearly and correctly justfying my answers, anything I can improve on or explain better? 2) Are my ...
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114 views

Division of large numbers

Euclidean Algorithm Versus Horner Algorithm. I have come across this problem and I do not manage to find a good example for it. For all the numbers I pick there is no loss. Give an example of a ...
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86 views

Finding all positive integers $m,n$ such that $\frac{n^3+1}{mn-1}$ is an integer

Determine all ordered pairs $(m,n)$ of positive integers such that $\dfrac{n^3+1}{mn-1}$ is an integer. My work: $$\frac{n^3(m^3+1)}{mn-1}=\frac{(mn)^3-1}{mn-1}+\frac{n^3+1}{mn-1}.$$ Since, ...
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75 views

GCD among all possible sudoku matrix determinants

Today I came across an interesting question Consider a completely filled Sudoku, written as a $9 \times 9$ matrix. Show that the determinant of this matrix is divisible by $405$. The solution ...
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0answers
59 views

Is it possible to find the least common divisor of a two numbers that are not relatively prime in polynomial time?

As the question states: Is it possible to find the least common divisor of two number that are not relatively prime in polynomial time? If so, how? Thanks!
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72 views

Prove that we always have $ 2n \mid \varphi(m^n+p^n) $

For each $ a ∈ \Bbb N^*$, denoted by $\varphi (a) $ is the number of positive integers not exceeding $a$ and coprime to $a$. Let $n, m, p ∈ \Bbb N^*, m \ne p$. Prove that we always have $2n \mid ...
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34 views

Computability of division of large numbers

What is the largest computable mathematical division in terms of the number of digits that can be handled by a typical desktop computer using the best available big number libraries, assuming input is ...
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0answers
160 views

K.K.T. conditions, Lagrangian gradient not defined for zero.

When I write the K.K.T. conditions for the problem I have, I get the following expression for the gradient of the Lagrangian: $$\frac{\partial \mathcal{L}}{\partial x} = - \frac{\sqrt{x} + ...
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158 views

Forcing and divisibility

I'm going to bring together a couple of seemingly unrelated questions that I've asked here. This may be silly. Or maybe not? Imagine that $n$ is some sort of infinitely large integer, and thus so ...
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0answers
12 views

Write a floored integer division in function of two divisions?

does exist any method to truncate the floored integer division for the sum of two numbers by a third one ? otherwise: if a%b + a%c < c $$\frac{a + b}{c} = \frac ac + \frac bc $$ if a%b + a%c >= ...