Questions tagged [divisibility]

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

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Parity of Euler's totient function

Let $S$ be a set of all numbers $k$ such that $(n, k) = 1, 1 \leq k \leq n.$ Of course, smallest element in $S$ is (by definition) $1$ and largest is $n - 1$ (since $\text{gcd}(n - 1, n) = 1$). In ...
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18 views

Polynomial Division Under Certain Remainders

Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-17$, the remainder is $14$, and when $P(x)$ is divided by $x-13$, the remainder is $6$. What is the remainder when $P(x)$ is divided ...
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1answer
38 views

Use Fermat's Little Theorem

Find a number $0 \leq a < 73$ with $a≡9^{794}\mod 73$. I know that $a$ and $73$ are relatively prime and $a^{72}≡1 \mod73$. But I couldn't use the theorem. Can someone help me please?
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How many solutions are there to the congruence

How many solutions are there to the congruence X^4 + 5X^3 + 4X^2 - 6X - 4 ≡ 0 (mod11) with 0 ≤X ≤11? I need to find that that if there are 4 solutions or there are fewer ...
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Diophantine equations that involve Gregory coefficients: a computational exercise

In this post, for integers $k\geq 1$, we denote the Gregory coefficients as $G_k$. Wikipedia has an article for Gregory coefficients, are known as reciprocal logarithmic numbers (I add this as ...
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The length of the algorithm [closed]

The "3n + 1 algorithm" works as follows. Start with any number n. If n is even, divide it by 2. If n is odd, replace it with 3n + 1 So, for example, if we start with 5, we get the list of numbers 5,...
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3answers
32 views

Divisibility of $a_{24}$ by 7. ($a_n=\underbrace{999\cdots9 }_{n \text{ times}})$

Question: By which number is $a_{24}$ divisible by? Where $a_n=\underbrace{999\cdots9 }_{n \text{ times}}$ The solution says the answer is $7$. Here's what is given: $$a_{24}=\underbrace{999\...
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2answers
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Find solutions of $m=\frac{n^2}{(n-m-1)\lambda+n}$ where $n,m,\lambda$ are postive integers,$1\le\lambda \le n-1$ and $m\mid n$.

I am considering the following equation $$m=\frac{n^2}{(n-m-1)\lambda+n}$$ where $n,m,\lambda$ are postive integers, $1\le\lambda \le n-1$ and $m\mid n$. If $m=n$, then $$\frac{n^2}{(n-n-1)\lambda+n}...
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Total ordering - Partially ordered set

A = {2,4,5,6,9,10,12,18,30,36,60,72} R={(a,b) | a divides b} I want to find a total order about partially ordered set(A,R). If there are multiple possible values, select a large number first. In ...
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Draw a hasse diagram about inverse R - divisibility

R = {(a,b) | a divides b} R is partially ordered set for set A. When I draw a hasse diagram about [inverse R], maybe I just change the top and bottom of the hasse diagram about [R]. Is it right?
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How to draw hasse diagram - divisibility - R and inverse R

I don't know how to draw the Hasse diagram for divisibility on the sets. A = {2,3,4,5,6,9,10} R is partially ordered set for set A. R = {(a,b) | a divides b} How to draw a hasse diagram about R and ...
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1answer
47 views

Unique solution of the equation $\prod_{i=1}^m(a_i+1)\prod_{i=1}^n(b_i)=\prod_{i=1}^m(a_i)\prod_{i=1}^n(b_i+1)$

Let $a_i$ be a sequence of $m$ distinct odd integers and $b_i$ a sequence of $n$ distinct odd integers. We have to prove that, $$\prod_{i=1}^m(a_i+1)\prod_{i=1}^n(b_i)=\prod_{i=1}^m(a_i)\prod_{i=1}^n(...
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Proof GCD(m,n) = GCD(m,n-m) [duplicate]

I need to prove that statement but I can't think of a way to do it. m and n are integers and m is different from 0.
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About the characterization of solutions of an equation that involves particular values of the Dedekind psi function

In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
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If $R$ is a UFD, $p(x)\in R[x]$ and $a/b$ is a root of $p(x)$ in the fraction field, then we have $p(x)=(bx-a)q(x)$ for $q(x)\in R[x]$.

Suppose $R$ is a UFD and $p(x)\in R[x]$ a polynomial of degree $\ge 1$. Suppose $\frac ab$ is in the fraction field $K$ of $R$, with $a$ and $b\in R$ and $\text{gcd}(a,b)=1$, and such it is a root ...
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4answers
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Last 3 digits of $2^{2017}$

Find the last three digits of $2^{2017}$ My approach: As $125 \times 8=1000$ we have the congruence modulo $$x \equiv 2^{2017}(mod \: 1000)$$ is equivalent to the equations $$x \equiv 2^{2017}(mod \:...
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52 views

How to check if a large number is divisible by a Prime Number? [closed]

How to check if a large number is divisible by a Prime Number? Are there some divisibility rules for this?
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1answer
38 views

Finding conditions such that $4b^2 > a^2 > 3b^2$ and $b \mid (a^2-1)$ imply $b=(a+1)/2$

Consider the set of odd positive integers $a$ and $b$ such that $4b^2 > a^2 > 3b^2$ and $b \mid (a^2-1)$. Brute-force computation suggests that $a=2b-1$ is the only solution for “most” such $b$,...
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1answer
35 views

Consider set $\mathbb{Z}[\sqrt{-5}] = \{a+b\sqrt{5}i : a,b \in \mathbb{Z} \}$ show that it is a ring [duplicate]

Consider set $\mathbb{Z}[\sqrt{-5}] = \{a+b\sqrt{5}i : a,b \in \mathbb{Z} \}$. My task is to show some features listed below: Show that $\mathbb{Z}[\sqrt{-5}]$ is a ring. I would like to show that ...
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2answers
28 views

Proof of divisibility by induction [duplicate]

I've recently come across a divisibility problem that I am unable to solve. I know that most of these types of problems have fairly straightforward proof-by-induction solutions -- but for this ...
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1answer
129 views

On a symmetric equation over the integer lattice that involves the Euler's totient function

I would like to know hints or a proof, or counterexamples, for the conjecture that I've stated in the Question below. I'm interested in this in an attempt to continue the study of a question that I've ...
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1answer
47 views

Find the number of incongruent solutions

Let $p$ be a prime number. Find the number of incongruent solutions of $$ x^{p^5}-x+p\equiv0\mod p^{2020}.$$ Let $f(x) = x^{p^5}-x+p$. Because of $f '(x)$ different from zero mod $p$. Then I say $$f(...
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41 views

For which $n\in N$ do we have $x^2+x+1|(x-1)^n -x^n-1$.

For which $n\in N$ do we have $$x^2+x+1|(x-1)^n -x^n-1$$ This question is from the book 'Problem Solving Strategies' Here is the solution Can anyone please explain the solution in detail? I am ...
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1answer
76 views

Prove any set S of three integers contains a pair $x\neq y$ such that $x^3y-xy^3$ is divisible by 10.

Prove any set S of three integers contains a pair $x\neq y$ such that $x^3y-xy^3$ is divisible by 10. My attempt was : By the division algorithm, every integer $n$ can be written as $n = 10q + r,$ ...
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1answer
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Find the smallest value $n$ such that there exists a non-empty subset of any set of n positive integers whose sum is divisible by 1001

Find the smallest value of $n$ such that for any set of $n$ positive integers, there exists a non-empty subset of the set whose sum is divisible by $1001$ This is sort of a follow up on my last post ...
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Proving that you can pick a group of numbers from a set of 4 natural numbers that divide 4 [duplicate]

Prove that for any set of $4$ natural numbers, it is possible to pick a group of numbers (can contain $1-4$ numbers) from the set such that the sum of the group is divisible by $4$. I tried to ...
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2answers
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$f(x)$ is monoic polynomial , prove that if $f(k), f(k + 1),…, f(k + p)$ is not divisible by $p + 1$, then $f(x) = 0$ has no rational solution. [duplicate]

Given a monic polynomial $f(x)$ of degree $n$ over $Z$ and $k, p \in N$ , prove that if none of the numbers $f(k), f(k + 1),..., f(k + p)$ is divisible by $p + 1$, then $f(x) = 0$ has no rational ...
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235 views

Related prime factors [closed]

Do there exist a positive integer $N$ and three primes $p_1<p_2<p_3$ such that the only prime factors of $N-p_1$ is $p_2$, the only prime factors of $N-p_2$ are $p_1,p_3$ and the only prime ...
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3answers
54 views

Prove that for all integers $n$, $3$ does not divide $n^2-5$ using modular arithmetic. [duplicate]

I am having trouble proving that for all integers $n,\ 3$ does not divide $n^2-5$ using modular arithmetic. I know that $3\not\mid n^2-5$ means $n^2\not\equiv 5\pmod 3$. But I'm not sure how to start ...
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1answer
50 views

Solutions to Diophantine equation $\frac{1}{n}+\frac{1}{p}=\frac{1}{N}$

For each prime $p$ there seems to be a uniqe solution $n=(p-1)p$ to the Diophantic equation $\frac{1}{n}+\frac{1}{p}=\frac{1}{N}$. Is that right and if so, how to prove the unicity? In spite of my ...
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2answers
103 views

Find the all positive integer solutions $(a,b)$ to $\frac{a^3+b^3}{ab+4}=2020$.

Find the all positive integer solutions of given equation $$\frac{a^3+b^3}{ab+4}=2020.$$ I find two possible solutions, namely $(1011,1009)$ and $(1009,1011)$, but the way I solve the equation was ...
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2answers
51 views

If $\alpha, \beta$ are the roots of the equation $x^2-(p+1)x+1=0.$ show that $\alpha^n + \beta^n$ is not divisible by $p$ $(p \ge3)$

Let $p \ge 3$ be an integer and $\alpha, \beta$ are the roots of the equation $x^2-(p+1)x+1=0.$ Using mathematical induction show that $\alpha^n + \beta^n$ (i) is an integer (ii) is not divisible by ...
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2answers
175 views

How to find k'th integer not divisible by n?

Although this was a programming question I want the mathematical intuition behind it. So we were given two numbers n and k. We were askd to find out k'th number not divisible by n. For example n=...
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3answers
89 views

When $ab/(a+b)$ is an integer, where $a,b$ are positive integers.

When $ab/(a+b)$ is an integer, where $a,b$ are positive integers? ...
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2answers
43 views

Prove that for odd $n > 1$ , $3^{n} + 1$ is not divisible by $n$ [duplicate]

Prove that for odd $n > 1$ , $3^{n} + 1$ is not divisible by $n$. There's a hint but I can't find any use for that. hint: If $a$ and $b$ are coprime with $m$ and $a^{x} \equiv b^{x}$ (mod $m$) ...
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1answer
48 views

I don't know if I'm correct.

Problem: Let $a,b\in\Bbb N$ with $a\cdot(a,b)=b\cdot[a,b]$ where (a,b) means the greatest common divisor of a,b and [a,b] means the smallest common multiple of a,b prove that there are an infinite ...
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2answers
57 views

A Divisibility Question

Consider $f(x) = x^5 + 2x^4 + x^3 + 2x^2 + x + 1 \in \mathbb{F}_3[x]$. I see that $f(x) = (x^2+x -1)\cdot(x^3 + x^2 + x - 1)$, both of which are irreducible / prime polyonimals (Recall that since $\...
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2answers
68 views

School-level problem on divisibility

I encountered the problem to show that there is an integer of the form $11111\ldots 11$ divisible by $2021$. It is easy to show that there is a number of the form $111 \ldots 11 \cdot 10^k$ ...
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105 views

Is it true or can it be proven that $\frac{n}{\varphi(n)} \geq \operatorname{rad}(n)$?

The title says it all: Is it true or can it be proven that $\frac{n}{\varphi(n)} \geq \operatorname{rad}(n)$? Here $n$ is a natural number, $\varphi(n)$ is the Euler-totient function of $n$, and $\...
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1answer
37 views

Long division answer differs to calculator answer

Can someone make sure I'm not going mad, I'm doing very simple long division: $271÷15$. I work it out as $18.06\bar 3$ but a calculator returns $18.0\bar 6$? What's going on here? Do I have a ...
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2answers
42 views

Solve equation with constrain on number of decimals?

Given this equation: 11 x + 102 y = 100 How can I find all possible solutions for which the following condition is satisfied: ...
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3answers
42 views

Find $a,b \in \mathbb{N}$ with $\mathrm{lcm}(a,b)=12\gcd(a,b)\,$ and $\,a\bmod b = 5$

Find $a,b \in\mathbb{N}$ such that: Remainder of $a$ divided by $b$ equals $5$ $\mathrm{lcm}(a,b)=12\gcd(a,b)$ I don't know how or where to start. The most similar problem I found was Find ...
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1answer
59 views

Find the value of the prime $p$ knowing that $p>5$ and $p│(3^{p+1} +5^{p-1} + 1)$

I am trying to find the value of $p$, a prime number of which we only know that $p>5$ and: $p│(3^{p+1} +5^{p-1} + 1)$ This is part of a collection of exercises regarding divisibility, Fermat's ...
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3answers
44 views

A function divisible by $p$

I want to ask about what is the intuition for making functions like these $4^n+2$ is divisible by $3$ $ 2^{4n+2}+3^{n+2}$ is divisible by $13$ And if so, how can I make my own ones? Thank you all ...
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0answers
21 views

2014 doors problem [duplicate]

$2014$ numerated doors. All doors are closed. Person $1$ opens all the doors. Person $2$ closes $2,4,6,8,...$ doors. Person $3$ changes (closes if door is open and vice versa) $3,6,9,12....
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1answer
77 views

There are two positive integers of the form $p-n^2$ such that one divides the other.

Let $p>3$ be a prime number. Consider the numbers of the form $p - n^2$ where $n = 1 , 2 , 3 , ... , \lfloor\sqrt{p}\rfloor$. Show that there exist two such numbers, $a$ and $b$, that $a | b$. I ...
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3answers
80 views

Dividing polynomial $f(x^3)$ by $x^2+x+1$

Given two polynomials with real coefficients $g(x)$ and $h(x)$. Additionally, $x^2+x+1$ is a factor of $f(x^3)= h(x^3)+xg(x^3)$. Prove that $x-1|h(x)$ and $x-1|g(x)$ I have tried to solve it by ...
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2answers
60 views

Proving that when $n$ is not a power of 2, $\exists k, 0 < k < n$ such that $\binom{n}{k}$ is odd

So I had a read over this question here which showed that any $n$ that is a power of two has it that $\binom{n}{k}$ is an even number for $0 < k < n$. Now I am wondering if for every other $n$,...
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0answers
32 views

Prove: $\gcd(a,b)=1\ $ and $\ n>ab-a-b \implies\exists\ x,y\ \epsilon \ \mathbb{N} \ /ax+by=n$ [duplicate]

Let be $a,b,n\ \in \ \mathbb{Z} \ $with $ a,b>1 $ We look at the equation $ax+by=n$ Prove that if $\gcd(a,b)=1\ $ and $\ n>ab-a-b \implies\exists\ x,y\ \in \ \mathbb{N} : ax+by=n$ I ...
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1answer
160 views

Help with this problem about a constructed number, that is from an arbitary n numbers, and that is divisible by a prime

Let $p$ be a prime number, and $n$ be an integer such that $n \geq p$. Let $a_1,...,a_n$ be arbitrary integers. Let $s_0 = 1$, and for every $k \ge 1$, let $$s_k=|\{B \subset \{1,2,...,n\} : p\mid\...

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