# Tagged Questions

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

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### if $p\mid a$ and $p\mid b$ then $p\mid \gcd(a,b)$

I would like to prove the following property : $$\forall (p,a,b)\in\mathbb{Z}^{3} \quad p\mid a \mbox{ and } p\mid b \implies p\mid \gcd(a,b)$$ Knowing that : Definition Given two natural ...
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### Which prime factors of $8^{8^8}+1$ are known?

We have the partial factorization $$8^{8^8}+1=(2^{2^{24}}+1)\cdot (2^{2^{25}}-2^{2^{24}}+1)$$ The first factor is $F_{24}$. It is composite, but no prime factor is known. A prime factor of the ...
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### Integers divide several solutions to Greatest Common Divisor equation

I'm not sure about the topic's correctness but my problem is following: Suppose $u_1,v_1$ and $u_2,v_2$ are two different solutions for $au_i + bv_i = 1$, then $a \mid v_2-v_1$ and $b\mid u_1-u_2$. ...
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### Find elements of a set that divide an expression.

I have to determine the elements of the following set: $A = \{x\in\ \mathbb Z \vert \sqrt[3]{\frac {7x + 2}{x+5}} \in \mathbb Z \}$ I know that $x+5 \not=0$ and $x+5$ must divide $7x + 2$ but I ...
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### $\frac{2abc}{(a+b-c)(b+c-a)(c+a-b)}$ a positive integer

Find all triplets $(a,b,c)$ of positive integers so that $\gcd(a,b,c)=1$ and $$\frac{2abc}{(a+b-c)(b+c-a)(c+a-b)}$$ is a positive integer. What I've done: first I looked with Mathematica for ...
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### Divide a number by two different numbers in combination

Similar to this question: How to divide a number by $2$ numbers? I would like to know how to divide a number (let's say 23) by two different numbers (say, 1.6 and 2.4) to find what combination of the ...
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### How to understand Apostol's proof of the irrationality of $\sqrt{n}$ if $n$ is not a perfect square?

Recently I am reading the textbook of Apostol, Mathematical Analysis, Second Edition. On page 7, there is a theorem 1.10: If $n$ is a positive integer with is not a perfect square, then $\sqrt{n}$ is ...
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### Number theory, prove that a prime number $p \mid 1$

Consider a prime number $p > 1$ and $a \in \mathbb{Z}$ and $p < a$. We know $p \mid a$, then $a = p.b$ for $b \in \mathbb{N}$. We also already know the congruence $a \equiv 1 (\text{mod } m)$ ...
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### If $k(a^2+mb^2) = c^2+md^2$, what can be said about the form of $k$?

Let $k,a,b,c,$ and $d$ be integers, and let $m \ge 2$ be a non-square integer, such that $$k(a^2+mb^2) = c^2+md^2.$$ QUESTIONS: What can be said about the form of $k$ with no further restrictions?...
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### Divisibility of Exponents

So I'm having trouble trying to show this, a,b and x are positive integers. If $a\mid b^x$, show that some factor $k$ of $a$ divides $b$. In other words, if a number $a$ divides a power, how can I ...
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### Proof using deductive reasoning

I need to deductively prove that the sum of cubes of $3$ consecutive natural numbers is divisible by $9$. I can prove deductively that they are divisible by $3$ but so far any combination I choose ...
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### Finding the integers

Find all integers $a,b,c$ with $1<a<b<c$ such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$. I cannot understand how to solve this. I would appreciate any help.
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### Is it divisible by $3^n$?

I need to prove that a number made up exactly $3^n$ $1$s and nothing else is a multiple of $3^n$. Well I think it is true that any number is a multiple of $3^n$ if the sum of its digits is. But I ...
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### Get first digits of a very large quotient

Is there a method to get the first $n$ digits of a quotient (ex. a thousand digit number divided by a 5 digit number) without dividing all the way through? I suppose long division until $n$ digits are ...
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### When does $a^b\mid b^a$

Let $a,b >1$ be integers. When does $a^b \mid b^a$? Certainly if this is true then $a\mid b$ by considering $a$'s prime factors. (not quite convinced). Also then if $b$ is prime then $a=b$. ...
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### How to prove $p^2 \mid \binom {2p} {p }-2$ for prime $p$?

How to prove $p^2 \mid \binom {2p} {p } -2$ for prime $p$? I have a hint: for $1 \le i \le p-1$, $p \mid \binom p i$. I cannot even start the proof. Please help.
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### Need an assistance with a specific step of a specific Division Algorithm proof

I'm trying to wrap my head around a Division Algorithm's proof. That is, Let $a, b \in \mathbb{Z}, a \neq 0$. Then there are unique $q,r \in \mathbb{Z}$ such that $b = qa + r, 0 \leq r < |a|$. ...
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### Prove that if $n^2$ is divided by 3, then also $n$ can also be divided by 3.

$n\in \Bbb N$ Prove that if $n^2$ is divided by 3, then also n can also be divided by 3. I started solving this by induction, but I'm not sure that I'm going in the right direction, any ...
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### Proof that if $3 \mid p^2$ then $3 \mid p$ [closed]

Course: Analysis (1st year course) Question: What does the formal proof of the following statement look like: if $3\mid p^2$ then $3\mid p$, with $p \in \Bbb Z$? Thank you. EDIT: I'd like to use ...
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### Show that $\gcd(80,8a^2+1)=1$

Show that $\gcd(80,8a^2+1)=1$ Let $\gcd(80,8a^2+1)=d$, then we have: $d|80a^2+10,80a^2\Rightarrow\ d|10$ So $d=1\ or\ 2\ or\ 5\ or\ 10$ Obviously $d$ can't be $2\ or\ 10$,but how can we show $d$ can't ...
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### Factorial question: number of trailing zeroes in 125! [duplicate]

How many zeros are after the last nonzero digit of 125! ? The answer is 31, but how do you solve it?
### Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$
Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$. This is what I have done so far: Let $d = \gcd(a,b)$. Then $d=ax+by$ for some $x,y$. Then $d^2 =(ax+by)^2 = a^2x^2 + 2axby+b^2y^2$. I am trying to create a ...
### Divisibility via Induction: $8^n\mid (4n)!$ [duplicate]
I have to prove by induction the following fact: Show that $(4n)!$ is divisible by $8^n$. My stab at the solution: I have a slightly bad feeling about this solution and I would like if someone ...