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

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$\operatorname{gcd}(ab,a+b)=1$ if $a$ and $b$ are relatively prime

I'm trying to show that if $\operatorname{gcd}(a,b) = 1$, then $\operatorname{gcd}(ab,a+b)=1$. I've tried to use the gcd properties: $$\operatorname{gcd}(a,b)=1 \implies ...
9
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1answer
110 views

For what integers $n$ is this divisibility statement true?

The statement being $$n^2 + 2 \mid 2014n + 2$$ The answer is $n = -2, 0, 1, 2014$. Don't know how to arrive at this answer without using comp sci. (Using the compsci answer, we can restrict the ...
4
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1answer
60 views

Show that each integer of the form $a^2+b^2$ has all the factors of this form, where $(a, b)$ are distinct integers and relatively prime

Show that each integer of the form $a^2+b^2$ has all the factors of this form, where $(a, b)$ are distinct integers and relatively prime Progress If $a^2+b^2$ is prime then it is already proved, ...
3
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2answers
145 views

Fastest way to find if a given number is prime

Given a random number, what would be the quickest possible way of finding out whether it was prime? Obviously, one could just iterate through the number in order to see if it was divisible by ...
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1answer
38 views

prove by contradiction that $ax+by=c$ has no integer solutions if $c$ does not divide into $\gcd (a, b)$

Prove by contradiction that (the diophantine equation) $ax+by=c$ has no integer solutions if $c$ does not divide into $\gcd (a, b)$. Here is what I did: lets assume $c$ divides into $\gcd (a, b)$. ...
3
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2answers
45 views

Prove that the Diophantine equation $ax+by=c$ has no integer solutions if $\gcd(a,b)$ does not divide $c$

Prove that the Diophantine equation $ax+by=c$ has no integer solutions if $\gcd(a,b)$ does not divide $c$ there was a hint which is use use contradiction.
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1answer
41 views

How prove $n|2^{\frac{n(n-1)}{2}}\cdot (2-1)(2^2-1)(2^3-1)\cdots (2^{n-1}-1)$

Question: Today, when I solve other problem, I found this follow interesting result $$n\mid\left(2^{\frac{n(n-1)}{2}}\cdot (2-1)(2^2-1)(2^3-1)\cdots (2^{n-1}-1)\right),n\ge 1$$ It is clear ...
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1answer
46 views

If $n$ is any positive integer whose last digit is $5$, then $5$ divides $n$

Prove that if n is any positive integer whose last digit is a 5, then 5|n Therefore, n is going to be 5, 15, 25, 35 etc ... b∣a states that 'b divides a' and we know that 5∣5, 5∣15, 5∣25, 5∣35 ...
1
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1answer
246 views

Is there a simple algorithm I can use for this?

if I were asked to find all integers between 1 and 100 that leave remainder 3 on division by 5 and leave remainder 4 on division by 7, how would I go about this? It seems like such a simple question ...
5
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3answers
97 views

If $n=3^{2^k}-2^{2^k}$, then $n\mid 3^{n-1}-2^{n-1}$

Let $k \in \mathbb{N}$ and let $n=3^{2^k}-2^{2^k}$. Show that $$n\mid 3^{n-1}-2^{n-1}.$$ I have no idea how to prove this. Any suggestions?
3
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1answer
333 views

Choose a k-subset such that its elements 's gcd is maximal

Given $n$ positive integer and a positive integer k. How to find a subset of size k such that its elements 's gcd is maximal (just give the maximum value of gcd is okay). Example: Give $3$ integers ...
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4answers
46 views

Evaluating the greatest common divisor of pairs $(a, a^2)$, $(a, a+1)$, $(a, a+2)$

I have a homework question which i'm struggling with, i would be interested in what method i should use to solve the following problems: ...
3
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4answers
68 views

Proving that (4-2/1)(4-2/2)…(4-2/n) in an integer.

We have to prove that $(4-2/1)(4-2/2)...(4-2/n)$ is an integer for $n\in\mathbb{N}$. Can we do this by induction? We prove for $n = 1$, which is trivial as $(4-2/1) = 2$ which is clearly an integer. ...
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2answers
36 views

Multiplicity of roots in finite fields of order prime. [closed]

I am having trouble with completing this question from last years exam (part a and d) Let p be a prime, and $f = x^5-1 = (X-1)(X^4+X^3+X^2+X+1) \in \mathbb{F}_p[X]$ Show: (a) if $p\neq 5$ then every ...
3
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2answers
73 views

Induction on GCD problem [duplicate]

This is a two part question Given $\gcd(a,b) = 1$ consider $$\gcd \left( \frac{a^n - b^n }{a-b}, a- b\right) $$ It appears that the value of this is always equal to $n$ or $1$. How to prove it? ...
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2answers
110 views

Why is $y^{x-1}-1$ divisible by $x$?

I wanted to know if there is a way to prove that $y^{x-1}-1$ is divisible by $x$. Where $x$ is a prime number and is not equal to $y$, and $y$ is any positive whole number besides $1$. For example, ...
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1answer
42 views

Check my proof : gcd(a,b)=1=gcd(x,y) => (xa,yb)=gcd(x,b) gcd(y,a)

Note: (x,y) means gcd(x,y) I managed to prove the next Proposition: Let $(a,b)=1=(x,y)$. Then $(x a,y b)=(x,b)(y,a)$. It can be easily be generalized for the case that $(a,b)\neq1$ and or ...
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4answers
95 views

Find a pair of integers $n,x$ such that $84 = nx + (n-1)n$ and $x$ is odd [closed]

I have a equation like this: $$84 = nx + (n-1)n$$ where, $x$ is odd. I need to find the fastest way to find a possible $n$ and $x$. (In this case: $n = 6, x = 9$) Edit: Maybe the background ...
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3answers
41 views

Easy question but tough judgement -Are both boundary points included in a time interval as in this question

So the question is - Six bells commence tolling together at intervals of 2,4,6,8,10 and 12 seconds respectively In thirty minutes how many times do they toll together? (A) 16 (B)15 This question is ...
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4answers
39 views

If GCD of x and y is G then GCD of x and x+y is also G. but how to prove it? [closed]

If GCD of x and y is G then GCD of x and x+y is also G but how to prove it?
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0answers
43 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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1answer
37 views

Length of smallest repunits divisible by primes

I want to prove this statement from Wikipedia: It was found very early on that for any prime p greater than 5, the period of the decimal expansion of 1/p is equal to the length of the smallest ...
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3answers
115 views

Number theory proofs regarding gcd's

How would you prove if $ad-bc = 1$, then $(a+c,b+d)=1$
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3answers
242 views

Diophantine equation $ax + by = c$ has an integer solution $x_0, y_0$ if and only if $\gcd(a,b)|c$

Let $a,b,c$ be positive integers. Verify that Diophantine equation $ax + by = c$ has integer solution $x_0, y_0$ if and only if $GCD(a,b)|c$. Attempt Diophantine $ax + by = c$ has integer solution ...
0
votes
1answer
55 views

Generalized Fibonacci Sequence

I'm having trouble with a problem I encountered while studying Number Theory. This problem comes from the book Number Theory by George E. Andrews. It defines a generalized Fibonacci sequence $F_1$, ...
2
votes
2answers
169 views

Divisibility for 7

I have seen other criteria for divisibility by 7. Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. 323) is interesting, but could not prove it. Let $n = ...
3
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5answers
121 views

How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?

I've been asked to show that: $n(n+1)(2n+1) \equiv 0 \pmod 6$ I found in a previous question that: $n(n+1)$ was divisible by $2$ and resulted in an even number e.g $n(n+1) \equiv 0 \pmod 2$ so I ...
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2answers
40 views

Solution for trinomial divided by binomial equation

I have the following equation to solve. I know that the answer is -5, I made several attempts at this, and arrive at a different answer. My first thought was to factor out the trinomial, but that ...
1
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3answers
85 views

Prove that $17$ divides $9a + 5b$

So, according to the book, for all $a, b, c$ that are elements of integers, it holds that $a|b$ implies $a|bx$ for all $x$ that is an element of integers. In other words it works for all ARBITRARY $x$ ...
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3answers
80 views

If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$

As stated in the title, the problem to prove is Let $a,b,c \in \mathbb{Z}$. If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$. I think I've proved it, but I would like a second opinion. Here ...
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1answer
42 views

Polynomial and its derivative have a common factor?

When is $gcd(p(x),p'(x))\ne 1$ where $p(x)$ is a polynomial? That is when does the derivative of a polynomial and the polynomial has a common factor? By when i mean some condition for the ...
4
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4answers
207 views

Inequalities, when does the sign change here?

I have encountered a problem with inequalities. I have been looking at examples provided by two websites which 'solve' inequalities, however when I try using my own method, the extremely simple ...
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0answers
45 views

Prove that $l = k/\gcd(m,k)$.

Suppose $ml = kt$ where $t$ is an integer and $m<k.$ $\implies k~|~ml$ $~~~~~$and $~~~~~$ $1 \leq \gcd(m,k) \leq m$ $\implies \dfrac{k}{\gcd(m,k)}~\Big|~\left(\dfrac{m}{\gcd(m,k)}\right)l$ What ...
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2answers
102 views

Divisibility crieteria

This is a follow-up question. The problem is: Given two natural numbers, $m$ and $n$, and $n \vert m^2$. Find necessary and sufficient conditions for $n \vert m$. Here are what I find: ...
3
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2answers
105 views

Ring such that $q^2\mid p^2$ does not imply $q\mid p$?

Let $R$ be a commutative ring with $1$ and suppose $q^2\mid p^2,$ for $p,q \in R$. Unless $R$ is a UFD, I don't believe I can conclude that $q\mid p,$ but I would like to know a concrete ...
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5answers
63 views

Direct proof using modular arithmetic

Give a direct proof of $8\mid (3^n + 5^n)$ for all odd natural numbers. I know how to prove this by induction, I am not sure how to go about it using a direct proof. I would start by saying that ...
2
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0answers
29 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 $p^\alpha$, ...
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1answer
85 views

Divisibility of huge numbers

Please help me to solve my homework ;) Prove that for any positive integer $n$ a square of rather big number divides even more huge number: $${\LARGE \left.\underbrace{33\dots 3}_{1\underbrace{00\dots ...
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0answers
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$U(n) =$ 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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2answers
40 views

Equation Theory: A polynomial with specific remainders when divided by specific divisors. What is the remainder when divided by BOTH divisors

Again, for my Equation Theory class, I have the subject question.$p(x)$ has a remainder of 3 when divided by $x-1$ and a remainder of 5 when divided by $x-3$. What is the remainder when $p(x)$ is ...
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1answer
65 views

Parity of number of factors up to a bound?

Consider $b,n\in\mathbb{N}$ where $b\leq n$. We want to find the parity (ie. odd or even) of the number of divisors of $n$ that are $\leq b$. The question is to find a fast algorithm to find that ...
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1answer
59 views

Question in elementary number theory

I have a question. Suppose that $a$ and $b$ are two natural numbers so that $ a<b$ and $ a\nmid b$. Put $ d=ka$, where $ k\not=0,1,t\dfrac{b}{\gcd(a,b)}$, for $ t\geq 1$. I want to prove that $ ...
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5answers
44 views

How do you prove this divisibility?

If $n$ is any natural number, prove that $3\mid 2^{2^n}-1$ is true. I can't find out how to do it. Thanks.
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1answer
71 views

Characterizing the primes which don't divide any Pell-Lucas number(s)

For integer $n$, let $P_n$ be a Pell number, and $Q_n$ its companion. Is there a characterization of the prime numbers $p$ which don't divide any $Q_n$? By brute-force search, I found that this ...
3
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1answer
106 views

Probability that two random integers have only one prime factor in common

The probability that two integers picked at random are relatively prime is known to be $1/\zeta{(2)}=6/\pi^2\approx0.607927...$. Generalizing, the probability that $n$ random integers have $\gcd=1$ is ...
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3answers
32 views

Help with understanding definition of divisibility in this case.

I have a proof that shows that if $5 \mid xy$ then $5 \mid x$ or $5 \mid y$. It's pretty clear to me that I can just say that suppose $5 \mid x$, then $x=5a$, where $a$ is an integer. then $xy = ...
3
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5answers
911 views

The sum of three consecutive cubes numbers produces 9 multiple

I want to prove that $n^3 + (n+1)^3 + (n+2)^3$ is always a $9$ multiple I used induction by the way. I reach this equation: $(n+1)^3 + (n+2)^3 + (n+3)^3$ But is a lot of time to calculate each ...
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0answers
34 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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1answer
34 views

How to get all divisors of an integer using only pen & paper

Is there any fast approach to get all divisors of an integer by only using pen & paper?
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0answers
75 views

Counting maximum moves

Given two arrays, each of size N denoted by A1,A2...AN and B1,B2...BN. Let us maintain two sets S1 and S2 which are empty initially. In one move ,Pick a pair of indexes (i, j) such that : ...