# Questions tagged [divisibility]

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

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### Find $100$th number $k$ such that there is no $n$ for which $n$! ends in $k$ zeroes.

$24! = 620,448,401,733,239,439,360,000$ ends in four zeroes, and $25! = 15,511,210,043,330,985,984,000,000$ ends in six zeroes. Thus, there is no integer $n$ such that $n!$ ends in exactly five zeroes....
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### Divisibility of $\frac{3^n - 1}{2}$ by $2^k$ [duplicate]

I would like to find the greatest k such that $p=\frac{3^n - 1}{2}$ is divisible by $2^k$. Since $p$ is the repunit number in base 3 it is already clear that if $n$ is even, $p$ would be divisible by ...
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### Are there infinite many primes $\ p\$ that cannot divide $\ 3^n+5^n+7^n\$?

Let $\ M\$ be the set of the prime numbers $\ p\$ such that $\ p\nmid 3^n+5^n+7^n\$ for every positive integer $\ n\$ , in short the set of the prime numbers that cannot divide $\ 3^n+5^n+7^n\$. ...
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### When is $n = \frac{a_{1}x^{2} + a_{2}x + a_{3}}{a_{4}x + a_{5}}$ an integer where $a_{i}$ are incredibly large integers? [closed]

I have the equation with the following form: $$n = \frac{a_{1}x^{2} + a_{2}x + a_{3}}{a_{4}x + a_{5}}$$ Where $a_{i}$ are incredibly large (e.g 1000 digits) but unrelated (not part of sequence or ...
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### Finding the maximum $k$ such that $(7!)!$ is divisible by $(7!)^{k!}\cdot(6!)!$

If $(7!)!$ is divisible by $(7!)^{k!}\cdot(6!)!$, then what is the maximum value of $k$? At first glance I couldnt think of anything except Legendre's formula for calculating powers of a prime in a ...
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### Is $\frac{3^n+5^n+7^n}{15}$ only prime if $n$ is prime?

Let $f(n)=3^n+5^n+7^n$ It is easy to show that $\ 15\mid f(n)\$ if and only if $\ n\$ is odd. I searched for prime numbers of the form $g(n):=\frac{3^n+5^n+7^n}{15}$ with odd $n$ and found the ...
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1 vote
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### How to understand the fast divisibility check for a double-width dividend?

I found this algorithm from the GNU factor utility. Given a double width dividend $n=n_1B + n_0$ and a single width odd divisor $d$, where $n_1, n_0, d < B=2^w, 2\nmid d$. Then with the precomputed ...
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### Prove that if $a \mid c$ and $b \mid c$, then $ab \mid c^2$ [duplicate]

I am trying to solve this problem, but I am unsure if my proof if sufficient or not. Anyways, here is what I have tried: So, by using the definition of "divides" I get: (i): If $a \mid c$, ...
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1 vote
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### Prove that $p_1\cdots p_k - 1$ and $p_1\cdots p_l - 1$ are coprime and that $p_1\cdots p_k + 1$ and $p_1\cdots p_l+1$ are coprime.

Let $p_n$ denote the nth prime and suppose $l\neq k$. Prove or disprove that $p_1\cdots p_k - 1$ and $p_1\cdots p_l - 1$ are coprime. Prove or disprove that $p_1\cdots p_k + 1$ and $p_1\cdots p_l+1$ ...
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### Is there a simple test for divisibility by seventeen in base-twelve? [duplicate]

I am investigating math in the dozenal (a.k.a. duodecimal, base-twelve) system. As part of this, I am compiling a list of tests for divisibility. (All numbers in this post are dozenal, not decimal, ...
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1 vote
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### On Carmichael function and aliquot parts of odd perfect numbers

We denote as $N$ an odd perfect number, and $d\mid N$ one of its divisors. We denote the Carmichael function as $\lambda(x)$, Wikipedia has the article Carmichael function dedicated to this number ...
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### Is there a simple test for divisibility by sixteen in base-twelve?

I am investigating math in the dozenal (a.k.a. duodecimal, base-twelve) system. As part of this, I am compiling a list of tests for divisibility. (All numbers in this post are dozenal, not decimal, ...
• 751