Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [divisibility]

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

If $a$ and $b$ are integers, $a$ divides $b$ if $b=ca$ for some integer $c$. This is denoted $a\mid b$. It is usually studied in introductory courses in number theory, so add if appropriate.

A common notation used for the phrase "$a$ divides $b$" is $a|b$. It is also common to negate the notation by adding a slash like this: "$c$ does not divide $d$" written as $c\nmid d$. Note that the order is important: for example, $2|4$ but "$4\nmid 2$".

This notion can be generalized to any ring. The definition is the same: For two elements $a$ and $b$ of a commutative ring $R$, $a$ divides $b$ if $ac=b$ for some $c$ in $R$.

Divisibility in commutative rings corresponds exactly to containment the poset of principal ideals. That is, $a$ divides $b$ if and only if $aR\subseteq bR$. For commutative rings like principal ideal rings, this means that divisibility mirrors exactly the poset of all ideals of the ring.

The topics appropriate for this tag include, for example:

  • Questions about the relation $\mid$.
  • Questions about the GCD and LCM.

There are divisibility rule that is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits.

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Remainder Being negative or positive?

I'm trying to determine if I would have a negative or positive remainder. I'm writing a program that would divide $-326$ by $7$. The program gives me $-46$ with a remainder of $-4$, however when I checked some other results online I got that my…
Camilo
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How can I determine the smallest quotient for all general cases?

If i have a big number, like: $39486432$ or $485921157$ And they ask me, with what number when dividing it, is it left with less residue? If you take into account the exact divisors. I would not like to use brute force, is there a wonderful theorem…
ESCM
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How do i prove this

Determine all the positive integers for which $p>2$ so that $p-1$ divides $p+11$ and show that if $k$ is a positive integer : $$p+11=k(p-1) \iff (p-1)(k-1)=12$$ i know that for example if a,b and a,c then a,(b+c) because we can just replace b = Xa…
alexJoegan
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Is $d|b$ and $d|a$ true in $d = l \cdot a + k \cdot b$

There are $a,b \in \mathbb Z, a \neq 0$ and $d$ which is the smallest number in $\mathbb N$ for that $d = l \cdot a + k \cdot b$ is true. ($k,l \in \mathbb Z$) I think that $d|b$ and $d|a$ is true, but I can't find proof for it, can you?
M. Dou
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Find all positive integers $n$ such that $72|(10^n+8)$

We can see that $n\ge3$ by objection. How to prove this?
user300045
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Divisibillity discret mathmatic

Is it all integers n true that if 10 |n^(2) it holds 10|n? Is it all integers n true that if 9 | n^(2) it holds 9 | n? How do you prove this does i take different integers to get whole numbers?
user3704516
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The difference between 3 digit number and reversing its digit is always divisible by

The difference between a 3 digit number and a number fomed by reversing its digit is always divisible by? a.6 b.9 c.12 d.19 How to approach this question?
avk avk
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How many numbers are there in range 1 to 1000 which contains digits 2 and 3 and divisible 2 and 3?

How many numbers are there in range 1 to 1000 which contains digits 2 and 3 and divisible 2 and 3? I know the answer to find count of numbers in range 1 to 1000 which are divisble by 2 and 3. But the above question also asks that those numbers must…
Mike
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divisibility criterion for integer numbers using congruences

let be a positive integer written in the form $$ \sum_{n=0}^{k}a(n)10^{n} $$ my question is how can i deduce using mathematics if the number is divisible by 2 , 4 or another higher integer using congruences or another math theorem? here…
Jose Garcia
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Direct proof divisibility: Suppose $x$ is an integer such that $2 \cdot 3 \cdot 4 \cdot 5 \cdot x = 59 \cdot 58 \cdot 57 \cdot 56 \cdot 55$

Suppose $\,x\,$ is an integer such that $\,2 \cdot 3 \cdot 4 \cdot 5 \cdot x = 59 \cdot 58 \cdot 57 \cdot 56 \cdot 55.\,$ Does $\,59 \mid x$? Does $\,29 \mid x$? Does $\,118 \mid x$?
christinaqwer
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need help with equasion

Well. My computer has fritzed up and I'm having to perform some lenghy task, it's processing 20 files every 2 seconds, it's at 459000 of 854528 Roughly how long in seconds might it take? I've currently tried; 854528 - 45900 / 20 = 831578 Is…
James Terry
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if c|(a+b) and c|(a-b) then c|a?

My professor claims: If $c\mid(a+b)$ and $c\mid(a-b)$, then $c\mid a$. How is this correct? I can't prove it, all I achieved is proving that $c\mid(2a)$ which isn't the same.
Dan
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How do I prove that $ 10 | 4^{n + 2 }+ 5^{n + 2 }+ 4^{n + 3 }+ 5^{n + 3 }$

I have no idea how to prove this. I don't even have an idea where to start from, could someone drop a few hints?
Ognjen Ognjanovic
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Prove that integer $n$ is divisible by $65$

Can anybody help me with this problem? Prove that for every integer $n$, $$p(n) = n^{18} - n^{14} - n^6 + n^2$$ is divisible by $65$. Thanks!
Aleksandre Bregadze
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Prove that $m^3\cdot 3^{k+3}-m^2\cdot 3^{2k+3}+m\cdot 3^{k+2}$ is divisible by $m\cdot3^{k+2}$

Prove that $m^3\cdot 3^{k+3}-m^2\cdot 3^{2k+3}+m\cdot 3^{k+2}$ is divisible by $m\cdot3^{k+2}$ where $m$ is some integer. I have encountered this during a proof by induction problem. I am not sure how to approach this problem. Could someone please…
user815231
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