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

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basic word problem!

Find the smallest number by which $108$ must be multiplied to give a multiple of $80$.
1
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4answers
53 views

Is $\gcd(2^{2n}+1, 3)=1$?

Can any one prove that $2^{2n}+1$ and $3$ are relatively prime for any integer $n$? I tried with a Matlab program and computed this gcd upto $n= 25$. I got 1 for all of them. So I suppose that the ...
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2answers
38 views

If $n \in \mathbb N$, under what conditions are $n$ and $n+2$ relatively prime?

The Statement of the Problem: If $n \in \mathbb N$, under what conditions are $n$ and $n+2$ relatively prime? My Thoughts: I know that the answer is that $n$ must be odd. However, I'm not sure how ...
1
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2answers
53 views

Interesting $0, 1$ sequence of numbers,after $n>2, a_n$is composite.

Let us have a finite sequence with only $0$ and $1$ digit in our numbers(it can begin with $0$ too). $a_n$ is the number, which we get if we write our number $n$ times next to each other. Prove, that ...
4
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3answers
61 views

Prove that ${x^2+y^2=z^n}$ has a solution in $\mathbb{N}$ for all $n$ in $\mathbb{N}$

I am solving it by stating that $$x^2 +y^2 =c^2$$ represents a circle. And when $$c^2=z^n$$ then , it represents a system of concentric circles with radius varying as $z$ varies or $n$ varies. So, for ...
4
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1answer
41 views

Prove that if $p \mid a_1a_2 \ldots a_n$, then $p \mid a_j$ for some $j$ with $1 \le j \le n$

Let $p$ be a prime number and $a_1, a_2, \ldots, a_n$ be integers. Prove that if $p \mid a_1a_2 \ldots a_n$, then $p \mid a_j$ for some $j$ with $1 \leq j \leq n$. The hint was to use induction. ...
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2answers
23 views

GCD theory - gcd(x, y) = 1

Take $n + 1$ numbers out of $1, 2, ..., 2n$. Show that there will be two numbers $x, y$ so that $gcd(x, y) = 1$. What I've got is: Let $d=gcd(a,b)$; by definition there are integers $a′$ and $b′$ ...
2
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2answers
42 views

Suppose $p$ is a prime number and $a$ is an integer. Show that if $p \mid a^n$, then $p^n \mid a^n$ for any $n \geq 1$?

I know that if $p \mid a^n$, I can say $a^n = pr$ for some integer $r$, you can also conclude that $\gcd(p, a^n) = p$, but I'm not sure how to use that information if I even can to show that $p^n \mid ...
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0answers
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1
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2answers
15 views

If a is prime to b and y, b is prime to x then prove that ax+by is prime to ab [on hold]

I don't know how to write a formal proof but i am able to tell the result
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2answers
39 views
2
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1answer
21 views

Prove that if $a, b, n\in \mathbb{N}, n\geq2\longrightarrow \sqrt[\leftroot{-2}\uproot{2}n]{a}\in \mathbb{Q} \iff a=b^n$.

Prove that if $a, b, n\in \mathbb{N}, n\geq2\longrightarrow \sqrt[\leftroot{-2}\uproot{2}n]{a}\in \mathbb{Q} \iff a=b^n$. I'm at a complete loss here, I tried using the order of a prime function but ...
2
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1answer
25 views

Example of a domain where all irreducibles are primes and that is not a GCD domain

One has the following relations for a domain $R$: $R$ GCD domain $\Rightarrow$ All irreducible elements are prime $R$ PID $\Rightarrow$ $(R$ GCD domain $\land$ $R$ statisfies ACCP$)$ $R$ UFD ...
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0answers
73 views

Fermat's Little Theorem and prime divisors

Let $a,b\in\Bbb N$ and $a+b$ be an even number. Assume $a^2 - b^2 - a$ is an exact square, say $c^2$. Let $m = \frac {a+b}2$ and $n = \frac {a-b}2$. Then, $$(4m-1)(4n-1) = 4(4mn-m-n) + 1 = ...
2
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1answer
40 views

An expression with gcd and abs is transformed magically!

There's a problem to calculate $\sum^{n}_{i=1}\sum^{m}_{j=1}\frac{|i-j|}{\gcd(i,j)}$, whose tutorial gives the following transformation I really don't understand. ...
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2answers
75 views

Pythagorean Triples : Is every positive integer $\gt$ $2$ part of at least one Pythagorean triple?

I was doing some basic number theory problems from Rosen and came across this problem: Show that every positive integer $\gt$ $2$ is part of at least one ...
5
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1answer
91 views

Pythagorean Triples : Show that exactly one of $x$, $y$, and $z$ is divisible by $5$

I was doing some basic number theory problems from Rosen and came across this problem: Show that if $(x, y,z)$ is a primitive Pythagorean triple, then exactly one of $x$, $y$, and $z$ is divisible ...
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2answers
37 views

List the elements of a set [closed]

Consider the universal set $N$, $$A = \{m: m\ |\ 16\}$$ and $$B = \{n: n \le 16 \text{ and } n \equiv 17 \mod 3\}.$$ List the elements of A, list the elements of B.
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5answers
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Mathematical induction [duplicate]

Prove that $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ where $n$ is a nonnegative integer. I have seen many questions on this site that contain the answer to this problem and I already know the solution, ...
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1answer
35 views

Find two numbers, given their greatest common divisor and least common multiple [closed]

Highest common factor (HCF) of two numbers is $20$. Least common multiple (LCM) of the same two numbers is $420$. Both numbers are higher than $50$. Find the $2$ numbers. I used factorising trees ...
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0answers
24 views

Prove or disprove : if $a\mid bc$ and $\gcd (a,b) = 1$, then $a\mid c$ [duplicate]

If $a\mid bc$ and $\gcd (a,b) = 1$, then $a\mid c$. How do I prove this?
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1answer
37 views

Proving $k$ is divisible by $3$ iff the sum of the digits of $k$ is divisible by 3 [duplicate]

I am trying to prove that $k$ is divisible by $3$ iff the sum of the digits of $k$ is divisible by 3 for all $k \in Z$. I am not even sure what tags to use because I am not sure of right methods to ...
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1answer
37 views

Decide if there exist $a$ and $b \in \mathbb{Z}$ such that $a^2=2b^2$.

Decide if there exist $a$ and $b \in \mathbb{Z}$ such that $a^2=2b^2$. $a,b \neq 0$ We have to solve this kinds of problems using the order of a prime function: $v_p(a) \in \mathbb{Z}$ which tells ...
2
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3answers
73 views

$5 \nmid 2^{n}-1$ when $n$ is odd

I want to prove that $$5 \nmid 2^{n}-1$$ where $n$ is odd. I used Fermat's little theorem, which says $2^4 \equiv 1 \pmod 5$, because $n$ is odd then $4 \nmid n$ , so it is done. can you check it ...
6
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2answers
87 views

Prove or disprove : $a^3\mid b^2 \Rightarrow a\mid b$

I think it's true, because I can't see counterexamples. Here's a proof that I am not sure of: Let $p_1,p_2,\ldots, p_n$ be the prime factors of $a$ or $b$ \begin{eqnarray} a&=& ...
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3answers
40 views

Find $GCD(n^2+1,n+1)$

$GCD(n^2+1,n+1)$, $n\in \mathbb{N}$ What I did: $n^2+1=(n-1)(n+1) + 0$ So I thought $(n^2+1:n+1)=n+1$ But that doesn't seem to be the case: $n=2$ $n^2+1=5$ $n+1=3$ $GCD(5,3)=1$ Why is the ...
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2answers
13 views

Prove the order of the group homomorphism of an element divides the order of the element.

Let $\phi : G \rightarrow H$ be a group homomorphism. Prove $\forall g \in G$, the order of $\phi(g)$ divides $g$. I've gotten to the point where I've shown that if, $ord(\phi(g)) < ord(g)$ then ...
1
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2answers
49 views

For a primitive Pythagorean triple $(a, b, c)$, is it always true that $\gcd(a,b) = \gcd(b,c) = \gcd(a,c) = 1$?

Let $(a, b, c)$ be a primitive Pythagorean triple. I know that $\gcd(a,b,c) = 1$. Is it always true that $\gcd(a,b) = \gcd(b,c) = \gcd(a,c) = 1$?
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3answers
27 views

Binary remainder not equal to the decimal remainder

I am having a weird result. I am dividing the binary number $10101010100000$ by $10011$. In binary division. I get $R= 0100$ which is 4. However, If I consider the decimal representation of the ...
3
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3answers
57 views

Let $p$ be a prime. Why is ${p^mn \choose p^m}$, where $p \nmid n$, not divisible by $p$? [duplicate]

Let $p$ be a prime. Why is ${p^mn \choose p^m}$, where $p \nmid n$, not divisible by $p$? $${p^mn \choose p^m} = \frac{(p^mn)!}{p^m!(p^mn-p^m)!} = ...
1
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2answers
89 views

how $1/0.5$ is equal to $2$?

My question is how $1/0.5$ is equal to $2$. I am not asking the mathematical justification that $1/0.5=10/5=2$. I know all this. I just want to know how it is two... a lay man justification. ...
2
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1answer
38 views

How to show that $\mathbb Z+x \mathbb Q[x]$ is a GCD domain ?

How to show that $\mathbb Z+x \mathbb Q[x]$ is a Bezout domain , that is sum of two principal ideals is again a principal ideal ? Or at least , how to show that it is a GCD domain ? ( This will then ...
1
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1answer
33 views

Looking for an example of a GCD domain which is not UFD

I know that every UFD (unique factorization domain ) is a GCD domain i.e. g.c.d. of any two elements , not both zero , exists in the domain . I am looking for an example of a GCD domain which is not ...
2
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0answers
28 views

Polynomials and Divisibility Rule.

The question is this - If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $h(x)=xf(x^3)+x^2g(x^6)$ is divisible by $x^2+x+1$, then which of the following are true? 1. $f(1)=g(1)$ ...
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1answer
30 views

Finding a natural number $k>1$ such that $k$ divides $(26+35n)$ and $(3+7n)$

I am trying to find a natural number $k>1$ such that $k$ divides $(26+35n)$ and $k$ divides $(3+7n)$ for some integer $n$. I know that $(ka)=(26+35n)$ for some $a \in Z$ and $(kb)=(3+7n)$ for some ...
2
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0answers
21 views

Show that $\gcd(x_1,…,x_k,x_{k+1})=\gcd(\gcd(x_1,…,x_k),x_{k+1})$

I would really appreciate if you could check my proof. Thank you. Show that $\gcd(x_1,...,x_k,x_{k+1})=\gcd(\gcd(x_1,...,x_k),x_{k+1})$ Let $\gcd(x_1,...,x_k,x_{k+1})=a$. We first show that $a$ ...
1
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4answers
78 views

Prove that if $3|mn$, then $3|m$ or $3|n$

I am trying to prove this for integers $m$ and $n$. I tried to reach prove that $3|m$ by assuming that 3 does not divide $n$, but this is such a basic assumption of mine already that it is hard for ...
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1answer
42 views

If $m\mid n$ then $p^m-1\mid p^n-1$ [duplicate]

I know $m$ ,$n$ are two positive integer numbers such that $m\mid n$. If $p$ is a prime number, I want to show $p^m-1\mid p^n-1$.
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3answers
31 views

How to formally prove: if $d\mid da+b$, then $d\mid b$?

How would I formally prove that for the integers $a$, $b$, and $d$, if $d\mid da+b$, then $d\mid b$? Would a direct proof be the best option? If I do a direct proof I seem to get stuck pretty ...
0
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1answer
18 views

Divisibility in a certain ring and divisibility in integers.

Divisibility in the ring $\mathbb{Z}[x,y]$ implies divisibility in $\mathbb{Z}$ ? Let $P(x,y)=Q(x,y)\cdot R(x,y)$ with $P,Q,R$ polynomials with integer coefficients, evaluating in $(x,y)=(a,b)$ with ...
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3answers
34 views

Proving gcd($a,b$)lcm($a,b$) = $|ab|$

I was trying to prove that $$ dm = |ab|$$ where $d$ = gcd(a,b) and m = lcm(a,b). So I went about by saying that $a = p_1p_2...p_n$ where each $p_n$ is a prime. Same applies to $b = q_1q_2 ... q_c$. ...
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2answers
43 views

Find $a,b,c$ if $\gcd(a,b,c)=10$ and $\text{lcm}(a,b,c)=100$ [closed]

Find all natural numbers $a,b,c$, so that $\gcd(a,b,c)=10$ and $\text{lcm}(a,b,c)=100$. Any ideas how to approach this problem?
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0answers
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Generalisation of $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n\gcd(a,b)^{n-1},a-b)$ to $\gcd\left(\frac{a^n-b^n}{a-b},a^m-b^m\right)$?

We have the identity $$\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n\gcd(a,b)^{n-1},a-b).$$ (see here) This appears to be a quite useful result with various applications. I wonder whether there is ...
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4answers
72 views

Prove if $7\mid a^2+b^2 \longrightarrow 7\mid a$ and $ 7\mid b$

Prove if $7\mid a^2+b^2 \longrightarrow 7\mid a$ and $7\mid b$ What I did: I found the possible remainders for $a^2$ are $0, 1, 2$ and $4$. I think I should say $r_7(a^2)+r_7(b^2)$ can't equal any ...
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0answers
27 views

$\max{a_1}$ in $(x_n)_{n\in\mathbb{N}}$

Let be $x_1,x_2,x_3,\ldots,$ a sequence of positive integers. Suposse the folowing conditions are true for all $n\in\mathbb{N}$ $n|x_n$ $|x_n-x_{n+1}|\leq 4$ Find the maximun value of $x_1$ I ...
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0answers
32 views

Existence of non-coprime between an integer and an arithmetic sequence

Take two relatively prime numbers $m,n \in \mathbb{Z}$ (i.e. $gcd(m,n) = 1$) where $m \neq 1$. Show that: $$\forall a \in \mathbb{Z} \textrm{ s.t. } 0<a<n$$ $$\exists i \in \mathbb{Z^+} ...
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0answers
33 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

(Note: This has been cross-posted to MO.) A positive integer $N$ is said to be perfect if $\sigma(N) = 2N$, where $\sigma(x)$ is the sum of the divisors of $x$. An odd perfect number $N$ is said to ...
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2answers
34 views

A question related to the concept of being “relatively prime”

Suppose that I have $a, b, c, d \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers. If I have the equation $ab = 2cd$ and I know that $\gcd(a,c)=\gcd(c,d)=1$, then it follows that I ...
2
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2answers
52 views

If $n$ is a positive integer such that $2^n+n^2$ is a prime number , then is it true that $6|n-3$ ?

If $n$ is a positive integer such that $2^n+n^2$ is a prime number , then is it true that $6|n-3$ ? Trivially $n$ cannot be even , so this leaves us only with the possibilities $n \equiv1,3,5( \mod 6) ...
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0answers
50 views

Prove that $\{ ax+by\mid x,y\in\mathbb Z\} = \{ n(a,b) \mid n\in\mathbb Z\}$ [duplicate]

Prove the following proposition: Suppose $a,b$ are fixed integers. Then $\{ ax+by\mid x,y\in\mathbb Z\} = \{ n(a,b) \mid n\in\mathbb Z\}$.