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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Divisibility of cyclic sums

Lately I have been studying the divisibility of some cyclic sums, and I was wondering about the following Conjecture Let it be a set of distinct positive integers $S=\{x_1,x_2,...,x_n\}$ such that $2\leq{x_1}
Juan Moreno
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can $n$ have more than $\dfrac{n}{2} + 1$ divisors?

Let $n$ be a natural number. Can $n$ have more than $\dfrac{n}{2}+1$ divisors? (a) $3$ 3 = 1 * 3 (2 divisors) 3/2 + 1 = 2 2 <= 2 [OK] (b) $4$ 4 = 1 * 2 * 2 (3 divisors) 4/2 + 1 = 3 3 <= 3 [OK] (c) $6$ 6 = 1 * 2 * 3 (4 divisors) 6/2 + 1 = 4 3 <=…
Daniel
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how can I find all numbers dividable by multiple numbers between two numbers

How can I find all numbers dividable by let's say 5,7,13 that are 2007***
Rick Stark
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Number formed by entering digits row wise or column wise in calculator numpad is divisble by 3 exactly two times not more or less than that. Why?

If we form numbers by entering digits in row or column wise in calculator numpad the resulting number is divisible by 3 exactly two times not more or less than that. Lets consider this numpad 7 8 9 4 5 6 1 2 3 Rule 1 Pattern should be same while…
Vishal Kumar
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When an integer is disible by 17,19,23, or 41?

Let $n=a_m10^m++a_{m-1}10^{m-1}+\dots + a_{2}10^2+a_{1}10+a_0$ where $a_k$ are integers and $0\leq a_k \leq 9,k=0,1,\dots,m$ be the decimal representation of a positive integer $n$. Let $S=a_0+a_1+\dots + a_m, T=a_0-a_1+ \dots + (-1)^m a_m$.…
Md Kutubuddin Sardar
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Prove the divisibility test by $7,11,13$ for numbers more than six digits

Prove the divisibility test by $7,11,13$ for numbers more than six digits Attempt: We know that $7\cdot 11 \cdot 13 = 1001$. The for a six-digit number, for example, $120544$, we write it as $$ 120544 = 120120 + 424 = 120\cdot1001 + 424 $$ thus we…
Redsbefall
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Does Euclid's Lemma apply to squared numbers?

Euclid's lemma states that: if a prime $p$ divides the product $ab$ of two integers $a$ and $b$, then $p$ must divide at least one of those integers $a$ and $b$. My question is can $a=b$? or must they be different integers? If they can would I be…
Krzysztof Jaminski
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Describe the set of all good numbers

A natural number $k$ is considered good, if for each $n$ the number $1^k+2^k+\cdots+n^k$ is divisible by $1+2 +\cdots+n$. Describe the set of all good numbers (with proof).
user64370
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Divisibility of fourths by seven

I am otherwise very good in mathematics, but recently I came upon a problem that I just can't solve. Do you have any idea how to solve it? If $a^2 + b^2 + c^2$ is divisible by 7, prove that $a^4 + b^4 + c^4$ is divisible by 7 as well. Thanks!
Pygmalion
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Describe the Natural density of $p$ which divides natural numbers of the form $n^2+1$?

We want to find the numbers that divide natural numbers in the form of $n^2+1$ and solve for their natural density. Using Wolfram Mathematica, I found divisors from $n=0$ to $1000000$ and eliminated repeated divisors. Here is the…
Arbuja
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Prove that if $a\mid(b-1)$ and $a^2\mid(b^2-2b+4)$, then $a\mid12$.

Prove that if $a\mid(b-1)$ and $a^2\mid(b^2-2b+4)$, then $a\mid12$. I am not sure where to start for this question, any help would be greatly appreciated, thanks!
macy
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Divisibility of a number by $3$

I have to prove the following statement by induction: $P(n):5^{3n} + 2^{n+1}$ is a multiple of $3$ for all $n \in \mathbb{N}$ I started with the base case for $n=1$, which is true, and then, by taking $P(n)$ as true, $P(n+1)$ gives me $125 * 5^{3n}…
user600210
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Infinite sequence $2^{n}-3 (n=2,3,...)$ contains no term divisible by 65

Show that the infinite sequence $2^{n}-3 (n=2,3,...)$ contains infinitely many terms which are divisible by $5$ and infinitely many terms which are divisible by $13$, but no terms which are divisible by $65$ My attempt at this:- By Fermat's…
user612946
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Prove that number $1^{2005}+2^{2005}+\cdots+n^{2005}$ is not divisible by number $n+2$

Prove that number $1^{2005}+2^{2005}+\cdots+n^{2005}$ is not divisible by number $n+2$ for every $n\in \mathbb N$ I have solution $2(1^{2005}+2^{2005}+ \cdots…
Marko Škorić
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Moving divisions to the opposite side

The question I have to answer is: $\frac {4V_3} 2 = 108V$, where I need to get $V_3$ on its own. When moving the 4 and the 2 to the other side will it be: $(108 \cdot 2)/4$ or $(108/4) \cdot 2 $
Daniel Holmes
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