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

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Contest Problem - Divisibility

Find all ordered pairs (x, y) of positive integers x, y such that $x+y$ divides 2014 and (simultaneously) $x^yy^x$ divides $(x+y)^{(x+y)}$ . This is a contest problem from U Tenn, FERMAT contest. My ...
6
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11answers
227 views

Prove that $n^2(n^2+1)(n^2-1)$ is a multiple of $5$ for any integer $n$. [closed]

Prove that $n^2(n^2+1)(n^2-1)$ is a multiple of $5$ for any integer $n$. I was thinking of using induction, but wasn't really sure how to do it.
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8answers
213 views

Why is $n^2+4$ never divisible by $3$? [duplicate]

Can somebody please explain why $n^2+4$ is never divisible by $3$? I know there is an example with $n^2+1$, however a $4$ can be broken down to $3+1$, and factor out a three, which would be divisible ...
2
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1answer
31 views

Proving divisibility of $\sum\limits_{r=1}^{p-1} {r^{p^n}}$ by p.

Let $p>2$ be an odd number and let $n$ be a positive integer. Prove that $p$ divides $${\sum\limits_{r=1}^{p-1}{r^{p^n}}}$$ My Proof: From multinomial expansion, we know that $${(1 + 2 + 3 + ... + ...
2
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2answers
50 views

Properties of Greatest Common Divisors

I really want to have help verify these properties of GCD: Let $s,t \in \mathbb{Z}$ and $m,n$ be positive integers with $m\mid n$. If $\gcd(t,n)\mid\gcd(s,n)$, then $\gcd(t,m)\mid\gcd(s,m)$. If ...
0
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2answers
47 views

divisibility of complex numbers

I want to show that $(a + bi)|(c + di)$ is equivalent to the statement that the ordinary integers $(a^2 + b^2)|(ac + bd)$ and $(a^2 + b^2)|(-ad + bc)$. I also want to show that $(a + bi)|(c + di) ...
4
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1answer
25 views

Square Roots: Variables with Exponents.

Alright, so let me get this straight: $\sqrt{x^2} = |x|$ $\sqrt{x^3} = x\sqrt{x}$ $\sqrt{x^4} = x^2$ $\sqrt{x^6} = |x^3|$ Are these correct?
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4answers
40 views

Question on modulus

Is $x|y$ the same as $x \equiv 0\! \mod\!{y}$ ? If not then how should it be written?
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2answers
23 views

Find the largest size of squares that can pave a given rectangle

The floor of a hall 252cm long, 162cm wide is paved with equal squares. Find the largest size of marble and number required, if only whole marbles are used. See the attempted solution posted as ...
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0answers
17 views

$x\in\mathcal{O}_K$ can be written as product of irreducible elements

Lemma: Every element $x\neq 0$ of $\mathcal{O}_K$ with ($K$ an arbitrary number field) can be written as a product of irreducible elements. Proof: We prove this lemma by complete induction on ...
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1answer
102 views

Is there a fast divisibility check for a fixed divisor?

Is there a fast algorithm to check if $d \mid n$ is true for varying $n$, if divisor $d$ is fixed? Variable $n$ is a $w$-bit binary integer, $d$ is an integer constant.
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4answers
169 views

Divisibility of $6^{2^n}+ 8^{2^n} +12^{2^n}+14^{2^n}+16^{2^n}+18^{2^n} +24^{2^n} +28^{2^n}+42^{2^n}$

Prove or disprove that for all natural $n$ $$6^{2^n}+ 8^{2^n} +12^{2^n}+14^{2^n}+16^{2^n}+18^{2^n} +24^{2^n} +28^{2^n}+42^{2^n}$$ is divisible by $259$. I tried to apply mathematical induction, but ...
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4answers
63 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
44 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 ...
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2answers
61 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
66 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
39 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 $b′$ ...
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2answers
47 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 ...
2
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1answer
23 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
36 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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1answer
85 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
45 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. ...
6
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2answers
106 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 ...
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1answer
115 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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5answers
100 views

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, ...
2
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1answer
46 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
75 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
127 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
42 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
18 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 ...
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2answers
55 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
28 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 ...
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3answers
58 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)!} = ...
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2answers
166 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
55 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, the 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 ...
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1answer
58 views

Looking for an example of a GCD domain which is not a 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 ...
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1answer
60 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
31 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
votes
4answers
87 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 ...
0
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1answer
45 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
34 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
votes
1answer
25 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 ...
2
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3answers
52 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 $b = q_1q_2 ... q_c$. ...
5
votes
0answers
46 views

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 ...
4
votes
4answers
211 views

Odd divisibility induction proof

Prove that for odd $n>3$ $$64\ | \ n^4-18n^2+17$$ I checked that for $n=5$ it works. I think I need to assume that for $2n+1$ it holds and show that $2n+3$ also holds. Any ideas?
2
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4answers
73 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
38 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
40 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 ...