Tagged Questions

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

43 views

Induction proof, divisibility

I'm struggling with an induction problem here. I have to prove that $2^{2^n}- 6$ (two to the power of two to the power of $n$ minus six) is divisible by $10$. I already figured some steps and I ...
251 views

Greatest common Divisor of negative numbers

To find gcd of negative numbers we can convert it to positive number and then find out the gcd. Will it make any difference?
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Find all positive integers solutions such that $3^k$ divides $2^n-1$

How can I find all positive of $k$ and $n$ such that $$\frac {2^n-1}{3^k}$$ is an integer? I know that $$2^n-1\equiv 0\pmod 3$$ If $n=2p$ with $p$ integer , $$2^n-1\equiv 0\pmod 9$$ If $n=6p$, ...
479 views

Find the $\gcd(81,237)$ and express it as a linear combination of $81$ and $237.$

How are they finding the encircled part. I am trying my very best to understand it, but in vain.
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Find all $\displaystyle n \in \mathbb{Z}$ such that $\displaystyle k = \frac{1+4n}{5}, \qquad (k \in \mathbb{Z} )$

My question is rather general but I got stuck in that issue after trying to solve a trigonometric equation. After simplifying I got this: $$\sin \left(\frac{5x}{4}\right) + \cos x = 2$$ which is ...
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Prove that there exist $2015$ consecutive abundant numbers [closed]

A positive integer $N$ is called abundant if the sum of its divisors is greater than $N$: $\delta (N) >N$. My question is: Prove that there exists an integer: $k\in\mathbb N\setminus\{0\}$ ...
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Prove that if $p$ is prime greater than $3$ ,then: $p^2+2015$ is multiple of $24$?

Prove that if $p$ is prime number $(p >3)$, then the number $p^2+2015$ is multiple of $24$? Thank you for any help
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Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$

I'm currently reading Andreescu and Andrica's Number Theory: Structures, examples and problems. Problem 1.1.7 states : Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$. The ...
3k views

What does “$x$ divides $y$” mean?

I need to negate the following sentence: "If for the integers $x, y, z$ we know that $x$ divides $y$ and $y$ divides $z$, then $x$ divides $z$." In this scenario, what does it mean for $x$ to ...
55 views

$\gcd(N, a)=\gcd(N, N-a)$ for positive integers $N$ and $a$?

If $\gcd(N, a)=1$, then we have $\gcd(N, N-a)=1$. More generally, can we have $\gcd(N, a)=\gcd(N, N-a)$ for positive integers $N$ and $a$? Thanks in advance.
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Sum of $m\leq 300$ such that if $2013m$ divides $n^{n}-1$, then $2013m$ also divides $n-1$

Find the sum of all the integers $m$ with $1≤m≤300$ such that for any integer $n$ with $n≥2$, if $2013m$ divides $n^{n}-1$, then $2013m$ also divides $n-1$. Unfortunately I cannot think of ...
I want to find the bezout coefficient for those 2 polynomials : $f = 1+x-x^2-x^4+x^5$ and $g = -1+x^2+x^3-x^6$ when I use the gcd function in sage the output is : ...