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

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3
votes
2answers
52 views

Prove that if $a \mid n$ then $a^2\mid (n + 1)(n − 1) + 1$

I have this review question for an exam and I was hoping someone can help me solve it: Prove that if $a \mid n$ then $a^2\mid (n + 1)(n − 1) + 1$ this is what I have so far, not sure if it is ...
0
votes
4answers
78 views

Let $a,b$ be relative integers such that $2a+3b$ is divisible by $11$. Prove that $a^2-5b^2$ is also divisible by $11$.

The divisibility for $11$ of $a^2 - 5b^2$ can be easily verified; in fact: $$a \equiv \frac {-3}{2}b \pmod {11}$$ therefore $$\frac {9}{4}\cdot b^2 - 5b^2 = 11(-\frac{b^2}{4}) \equiv 0 \pmod {11}.$$ ...
1
vote
9answers
88 views

Find integers $m$ and $n$ such that $14m+13n=7$.

The Problem: Find integers $m$ and $n$ such that $14m+13n=7$. Where I Am: I understand how to do this problem when the number on the RHS is $1$, and I understand how to get solutions for $m$ and ...
9
votes
0answers
118 views

Remainder of dividing $3^n$ by $2^n$.

I'd like to find the remainder of dividing $3^n$ by $2^n$, that is, I'd like to find value of $r$ in the expression $$3^n=q2^n+r,$$ where $q\in\mathbb{Z}$ and $0<r<2^n$. I know that it can be ...
1
vote
4answers
122 views

Why the $GCD$ of any two consecutive fibonnaci numbers is $1$?

Note: I've noticed that this answer was given in another question, but I merely want to know if the way I'm using could also give me a proof. I did the following: $$F_n=F_{n-1}+F_{n-2} \\ ...
1
vote
4answers
31 views

A={n∈ℤ: 6∣n and 8∣n} and B={n∈ℤ: 48∣n}. Is A⊆B?

A={n∈ℤ: 6∣n and 8∣n} and B={n∈ℤ: 48∣n}. Is A⊆B? Is B⊆A? I'm pretty sure that they are subsets of each other, because any n that 6 and 8 would both divide would have to be divisible by 6*8, but I'm ...
2
votes
5answers
192 views

Prove the existence or the non-existence of a couple of numbers ($n$,$m$) such that $n^2=m!$ [duplicate]

In recent days, while I was doing exercises on combinatorics, I thought if a number $m!$ could be a perfect square. I proved to demonstrate it through the prime factorization. My attempt: ...
4
votes
3answers
352 views

Prove the sum of any $n$ consecutive numbers is divisible by $n$ (when $n$ is odd).

Let $n \in \mathbb N$ be odd. Prove that the sum of any $n$ consecutive numbers is divisible by $n$. I started out with $s = x + (x + 1) + (x + 2) + … + (x + n) = kx + n.$ What I am interested in ...
-1
votes
3answers
59 views

show that $3^{(p-1)/2} +1$ is divisible by $p$ [closed]

let $n$ be an integer $>1$, and suppose that $p=2^n+1$ is a prime. Show that $3^{(p-1)/2} +1$ is divisible by $p$ (First show that $n$ must be even)
-1
votes
3answers
35 views

Prove that if $ r,p \in \Bbb{N} $, then $ \gcd(r,rp) = r $.

Problem. Prove that if $ r,p \in \Bbb{N} $, then $ \gcd(r,rp) = r $. I tried solving this. If $ \gcd(r,p) = 1 $, then $ \gcd(r,rp) = 1 \times r $. Is that right?
2
votes
1answer
23 views

Show that there exist $k$ and $r$ such that the given sum is divisible by $n$

Let $a_{1},\dots,a_{n}$ be integers. Show that there exist integers $k$ and $r$ such that the sum $$a_{k}+a_{k+1}+\dots+a_{k+r}$$ is divisible by $n$. I am unable to find the necessary way to solve ...
1
vote
1answer
24 views

Find GCD in Q[√3] assuming it is defined

How do I find the GCD of 24 and 49 in the integers of Q[√3], assuming that the GCD is defined?
-1
votes
2answers
130 views

Writing a GCD of three numbers as a linear combination

i know how to find the $\gcd(5,11,2^{2015}-1)$. but i can't seem to find the linear combination. do you find it the same way we find the linear combination of two integers.
1
vote
1answer
16 views

quadratic form polynomial divisibility vs. matrix pointwise multiplication.

Given matrix $V',W',Y'$ is of $d\times m (d\le m)$ ; column vector $c$ is of size $m$; $r_i, i=1,...,d$ are distinct; and each row of the matrix A is $A_i=(r_i^0 ... r_i^{d-1})$. So, A is of $d\times ...
3
votes
1answer
66 views

Find all $c\in\mathbb Z^+$ for which $\exists a,b\in\mathbb Z^+, a\neq b$ with $a+c\mid ab$ and $b+c\mid ab$

Find all $c\in\mathbb Z^+$ for which $\exists a,b\in\mathbb Z^+, a\neq b$ with $\begin{cases}a+c\mid ab\\b+c\mid ab\end{cases}$ For those $c$, prove only finitely many $(a,b)$ exist. ...
0
votes
2answers
315 views

Find the probability that an integer selected between 1 and 5000 is divisible by at least one of 3, 5 and 7

I'm having a hard time finding the solution. I can find integers that are divisible by only one of them, but there are many that are divisible by two of them. That's the problem. Find the probability ...
4
votes
5answers
112 views

Proof that $(n^7-n^3)(n^5+n^3)+n^{21}-n^{13}$ is a multiple of $3$.

I proved that $$(n^7-n^3)(n^5+n^3)+n^{21}-n^{13}$$ is a multiple of $3$ through the use of Little Fermat's theorem but i want to know if there exist other proofs(maybe for induction). How can I ...
1
vote
1answer
39 views

Analytical solution for $\max{x_1}$ 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 ...
1
vote
4answers
82 views

How find the fractional part of $5^{200}$ divided by $8$?

Finding the fractional part of $\frac{5^{200}}{8}$. I've had this problem given to me (we're learning the Binomial Theorem and all.) So obviously I thought I'd apply the binomial theorem to it, ...
1
vote
2answers
107 views

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 ...
4
votes
11answers
232 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.
4
votes
8answers
234 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 ...
4
votes
1answer
44 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
votes
2answers
70 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 ...
1
vote
2answers
65 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
votes
1answer
37 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?
0
votes
4answers
45 views

Question on modulus

Is $x|y$ the same as $x \equiv 0\! \mod\!{y}$ ? If not then how should it be written?
1
vote
2answers
29 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 ...
1
vote
0answers
19 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 ...
4
votes
1answer
159 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.
10
votes
4answers
179 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 ...
1
vote
4answers
66 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 ...
1
vote
2answers
58 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
vote
2answers
62 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
votes
3answers
67 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
votes
1answer
44 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. ...
1
vote
2answers
96 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
votes
2answers
51 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
votes
1answer
27 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 ...
3
votes
2answers
75 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 ...
1
vote
1answer
93 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
votes
1answer
48 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
votes
2answers
159 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 ...
6
votes
1answer
234 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 ...
0
votes
5answers
133 views

Mathematical induction: $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ [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
votes
1answer
72 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 ...
2
votes
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
votes
3answers
77 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
votes
2answers
136 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&=& ...
0
votes
3answers
58 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 ...