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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Prove that if $x$ is odd then $x^2 + (x + 2)^2$ is divisible by $2$ but not for $4$

This is what I tried: $$x^2 + x^2 + 4x + 4=2x^2 + 4(x + 1),$$ so it's divisible by $2$, since this expression is a sum of a multiple of $2$ and a multiple of $4$. Therefore, for the expression not to be a multiple of $4$, $2x^2$ can't be multiple of…
Vinicius Guimarães
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$XY$ minus $YX$ is always divisible by $3$

Why a $2$-digit decimal number minus the same number with the digits reversed is always divisible by $3$ ?
Michael Shenouda
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Denominator Identity Verification

Here is the given information for the following proof I'm about to begin: Suppose a, b, c, d, e and f are non-zero elements of field such that $$ \frac{a}{b} = \frac {c}{d} = \frac{e}{f}$$ I'm suppose to show the following identity is true,…
John W. Smith
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Number of $(p,q)$ in $\mathbb R^2$ such that $x^4+px^2+q$ is divisible by $x^2+px+q$

I have found four through my attempt, but apparently the answer is that there are five pairs. Mine were $(-1,0),(-1,1),(0,0),(0,1)$. What am I missing?
John Doe
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Divisibility by $8$ and $9$.

How many ordered pairs $(a,b)$ exist such that the four-digit number, $a04b$, is divisible by both $8$ and $9$? How should I approach this? (without modular arithmetic?)
user409878
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Prove or disprove: $a\mid(bc)$ if and only if $a\mid b$ and $a\mid c$

Prove $a\mid(bc)$ if and only if $a\mid b$ and $a\mid c$. My attempt is proving the converse first so if $a|b$ and $a|c$ then $a|bc$ So since $a\mid b$ and $a\mid c$, then $b=ax$ and $c=ay$ for some integers $x$ and $y$. So $bc=a(xy)$ therefore…
HighSchool15
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What is a niven number?

How do you work out if a number (214) is a niven number with a base n? but it cant be just to base 10 the number needs to be divisible by the sum of its digits when written in base n
sandra.williams
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Characterization of primitive roots modulo a prime

Let $p$ be a prime. Given a divisor $d$ of $p-1$, there always exist numbers with multiplicative order $d$ modulo $p$, and there are $\phi(d)$ such numbers. If for example $a$ has order $d$, then $a, a^2, ... , a^{d-1}$ are $d-1$ noncongruent…
D_S
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How to prove that $ab | \lceil \frac{n}{4} \rceil +c$?

Prove that for all positive integers $n \geq 17$ such that $\left\lfloor \frac{n}{4} \right\rfloor \equiv 0 \pmod 2$, there exists positive integers $a, b$ and an even positive integers $c$ such that the following are satisfied: $$ a+b = 4c-1$$ and…
Andrew
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$\forall n\in\mathbb N$ prove that at least one of the number $3^{3n}-2^{3n}$,$3^{3n}+2^{3n}$ is divisible by $35$.

$3^{3n}-2^{3n}=27^n-8^n=(27-8)(27^{n-1}+27^{n-2}\cdot 8+...+27^1\cdot8^{n-2}+8^{n-1})$ If $n$ is even, $3^{3n}+2^{3n}=27^n+8^n=(27+8)(27^{n-1}-27^{n-2}\cdot 8+...-27^1\cdot8^{n-2}+8^{n-1})$ If $n$ is even and a power of $2$, $3^{3n}+2^{3n}$ can't…
user300048
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Prove: $2730|(n^{13}-n),n\in\mathbb N$

$2730=2\cdot3\cdot5\cdot7\cdot13$ $n^{13}-n=n(n-1)(n+1)(n^2+1)(n^8+n^4+1)$ Divisibility by $2$ and $3$ follows from the product of two and three successive terms. Divisibility by $5$ follows from Fermat's little theorem: $$n^{5-1}\equiv1(\mod…
user300045
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Is $30$ a factor of $n$?

Let $n$ be an integer and assume that $30$ is a factor of $n^2$ and $15$ is a factor of $n$. Prove that $30$ is a factor of $n$. I tried testing some numbers, e.g., $n = 30$ clearly works since $15|30$ and $30|30^2$. Also $n = 60$ works. But, how…
user3184733
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Pythagorean triples $(a,b,c)$ and divisibility of $a$ or $b$ by $3$.

Call a triple of integers $(a, b, c)$ a Pythagorean triple if $a^2 + b^2 = c^2$ , i.e., if $a, b, c \in \mathbb{N^*}$ are the (measures of) sides of a right triangle. Examples of Pythagorean triples are (3, 4, 5), (5, 12, 13), (8, 15, 17) and…
user3704516
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Do we divide first or multiply first if we don't have any other information

Possible Duplicate: What is 48÷2(9+3)? 6/2*(1+2) is 1 or 9? order of operations division Apparently a very simple question but My question basically is, whether the answer of the following equation is 9 or 1? 6 / 2 ( 1 + 2 ) Or put another way,…
Aamir
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What is the remainder when 10,987,654 is divided by 2,103?

What is a simple way to solve this problem? I solve this problem by actually dividing $10,987,654$ by $2,103$, which should not be a simple way. What is the remainder when $10,987,654$ is divided by $2,103$?
learning
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