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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'Simple' multiple of an integer

Is there a special name in number theory for an integer a that is multiple of an integer m, but such that the quotient a/m is no longer divisible by m? Would it be a good idea to call such an m-multiple a SIMPLE (or maybe a PROPER) m-multiple? For…
frank
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Explanation of proof common divisor divides gcd needed

I have been given a proof that uses the gcd to show that there is no biggest prime. However (coming from a less mathematical background) I am having trouble actually understanding what it means and was wondering if someone could provide an…
mino
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How many solutions are there to the congruence

How many solutions are there to the congruence X^4 + 5X^3 + 4X^2 - 6X - 4 ≡ 0 (mod11) with 0 ≤X ≤11? I need to find that that if there are 4 solutions or there are fewer than 4 solutions? I saw this question while i was studying congeruences…
user792583
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Reduce the size of two numbers but keep their ratio

I have two numbers: 1536 and 2048, I would like to reduce these numbers to as close as 600 as possible while retaining their ratio. How can I achieve this in mathematics?
S-K'
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Prove that for odd $n > 1$ , $3^{n} + 1$ is not divisible by $n$

Prove that for odd $n > 1$ , $3^{n} + 1$ is not divisible by $n$. There's a hint but I can't find any use for that. hint: If $a$ and $b$ are coprime with $m$ and $a^{x} \equiv b^{x}$ (mod $m$) and $a^{y} \equiv b^{y}$ (mod $m$) then $a^{\gcd(x,y)}…
mathisfunny
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Is there a more efficient way to calculate "step" division?

I have a base value 16, I need to divide it by the ratio 1.067 6 times to reach the desired outcome value 10.84, I'm currently doing this via: 16 / 1.067 / 1.067 / 1.067 / 1.067 / 1.067 / 1.067 Is there a more efficient way to write this…
Tom
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Find all $n$ for which $19 \mid 10^n - 1$

For what values of $n$ (positive integer) does 19 divide $10^n-1$ evenly? The question arises in my fooling around with $2$-parasitic numbers. And I know this is true for $n = 18, 36, 54$ (by computing!) I expect this to be so for all multiples…
Michael Marcal
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Formula for moving object 1px when scaled down to 0.4824074074074074

This question comes within the context of web development but it is definitely a mathematical question IMO. As I drag my object around on the (x,y) axis. I am getting the coordinates returned to me in 1px increments i.e 1,2,3,4,5. The problem is my…
Michael Joseph Aubry
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Divisibility of Integers and TD

Is it true that if $3|x$ and $3|y$ and $3|z$ then $3|(x+y+z)?$ If yes how do I prove it? If not how do I work through a problem like this?
Johhny Smith
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Prove that the division of the square of an integer by 3 never yields 2 as the remainder.

I tried to conclude contradiction.I said n^2=2 (mod 3) . But I couldn't find a Contradiction.How Can I solve this question.
David
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Proving a | ($b^2$ + $c^2$) if a | b and a | c

To prove the statement above, I made this attempt: Let a, b, c ∈ ℤ, and a | b and a | c. Then ∃ j, k ∈ ℤ such that aj = b and ak = c. Then $b^2$ + $c^2$ = $(aj)^2$ + $(ak)^2$ = $a^2$($j^2$ + $k^2$). The problem is I need to show a(integer), not…
Val
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Can someone help splitting a range into equal parts

I have been stuck with a problem. I been presented a range from 10k to 3000k. I need to split this into ten sections. I go to do the math and things are not working out, can someone show me what i need to do an offer guidance because, i need to do…
Ben Madison
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$S= 2^{1} + 22^{11} + 222^{111} \cdots +22222222222^{11111111111}$. Remainder if $S$ is divided by $7?$ Prove that $S$ is not divisible by $5.$

Let, $S= 2^{1} + 22^{11} + 222^{111} \cdots +22222222222^{11111111111}$. What will be the remainder if $S$ is divided by $7$? Prove that $S$ is not divisible by $5.$ Attempt: Can the first part be solved by using the Chinese Remainder Theorem…
trombho
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Ways to prove a * b ^ g(n) + c * d ^ h(n) is divisible by e

I'm currently analyzing whether functions with the form $f(n) = a * b ^ {g(n)} + c * d ^ {h(n)}$ yield a result that is divisible by a number e for all $n \in \mathbb{N}$. Often I can show that this is true via induction. However, I've found some…
quant
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Find all $n$ with exactly $8$ divisors (counting $1$ and $n$ itself), the sum of which is equal to $684$.

Find all $n$ with exactly $8$ divisors (counting $1$ and $n$ itself), the sum of which is equal to $684$. I've written down two formulas from there https://en.wikipedia.org/wiki/Divisor_function?wprov=sfla1 and got stuck.
young
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