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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A 5 digit number is formed using the digits 0,1,2,3,4 and 5, probability it is divisible by 6?

A $5$ digit number is formed by using the digits $0,1,2,3,4$ and $5$ without repetition. The probability that the number is divisible by $6$ is? Answer : 18% I had doubts regarding this question but while writing my attempt here, I got the…
Arishta
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Prove that $n^5+n^4+1$ is composite for $n>1.$

Prove that $f(n)=n^5+n^4+1$ is composite for $n>1, n\in\mathbb{N}$. This problem appeared on a local mathematics competition, however it looks like there is no simple method to solve it. I tried multiplying it by $n+1$ or $n-1$ and then tried…
user333900
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A question about divisibility by $7$

There is a 6 digit number $abcdef$. Given that it is divisible by $7$, show that $abc-def$ is also divisible by $7$. Attempt at solution: $abcdef=10^5a + 10^4b + ... + f=7x$. Therefore, $7|a, 7|b, ...$. Hence, $abc=7m, def=7n$ and $7(m-n)$ must be…
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Number theory proof: Let $w=a+\sqrt{3}\cdot b$ where $w\neq 0$. Let $|N(w)| = 3^{a}\cdot k$, where $3\nmid k$...

Let $w$ be an extended integer (of the form $a+b\sqrt{3}$) with $w\neq 0$. Let $|N(w)| = 3^{a}\cdot k$, where $3\nmid k$ (See hw 4). Prove that there exists am extended integer $z$ such that $w = (\sqrt{3})^a\cdot z$ and $|N(z)| = k$. [Note- In the…
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Proving that $3^n+7^n+2$ is divisible by 12 for all $n\in\mathbb{N}$.

Can someone help me prove this? :( I have tried it multiple times but still cannot get to the answer. Prove by mathematical induction for $n$ an element of all positive integers that $3^n+7^n+2$ is divisible by 12.
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For how many numbers $X^2 \equiv X \mod 10^n$?

I've been looking through this post: Square of four digit number $a$ And wanted to see if there's a way to generalize the idea, in order to answer the following question: For how many numbers $X^2 \equiv X \mod 10^n$? Also, is $9376$ the only…
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if $2^n+1$ is prime, then n = $2^r$ and $(x - y) \mid(x^k - y^k$) only with natural numbers

I need to prove that if $2^n + 1$ is prime, then $n = 2^r$ for a natural number $r$. I do know how to prove it with the lemma $(x - y) \mid (x^k - y^k)$, but in order to prove it with this lemma, I need to substitute $y$ with $-1$ and my problem is,…
Joey
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Find all postive $m$ such $\gcd(m,2n!+1)=1,\forall n\in N^{+}$

Find all postive integer $m$,such $$\gcd(m,2n!+1)=1,\forall n\in N^{+}$$ when $n=1$ then we have $$\gcd(m,3)=1$$ when $n=2$,then we have $$\gcd(m,5)=1$$ when $n=3$,then we have $$\gcd(m,13)=1$$ when $n=4$, then we…
math110
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Prove that $\forall n \in \Bbb N, ~40^n n! \mid (5n)!$

I'm having trouble trying to solve this problem: Prove that $\forall n \in \Bbb N,~ 40^n n! \mid (5n)!$ I must be overlooking something simple because I can't go through it with induction. Base case $P(1)$ works. I want to see that it holds for…
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Find the smaller palindrome that is divisible by $3,5,11$

It seems to me that there are only two ways to solve the problem: First-Use divisibility rules to select last digits, and work my way from there Second-Write out all multiples of 165 since 3, 5, and 11 are relatively prime Both ways seem pretty…
Gerard L.
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Congruence equation.

Find all the $x$ for which there exists $y$ such that $x^2+y^2 \equiv x \mod xy$. I can write it as $k(xy)+x=x^2+y^2$ for some $k$. But I don't really know how to move on from here.
eremite
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is $0|0$ allowed??

since I can remember, I've been told that $0 / 0$ is not allowed because division by zero is not defined, but $0|0$ is okay because it means there exist a $k$ such a that $0 = k$ * $0$. and $k$ can be any number. but now, in a new book, I've read…
Fatemeh Karimi
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$\binom{3n}{n}$ divisible by $2016$

If $\dbinom{3n}{n}$ divisible by $2016$ then, what is smallest value of positive number of $n$? Notes: I can find $n=23$. $2016=2^5\cdot 3^2 \cdot 7$ We can give to $n=1,2,\dots ,22$ and $2016 \not| \dbinom{3n}{n}$. For $n=23$, $\dbinom{69}{23}$…
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General rule to determine if a binary number is divisible by a generic number

I always find myself doing tests with binary numbers (without a calculator, I'm now developing automatas) and I've always asked myself if there was a fast trick to check whether a generic number is divisible by another binary number. Let's say I…
Matt
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Divisibility Problem: How many switches remain in their initial position?

I have been working on this problem: There is a set of $1000$ switches, which are ordered in a row so that each switch is given a distinct rank from $1$ to $1000$. For example, the $i$-th switch refers to the switch given rank $i$. Each switch has…
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