# 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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### Factors/divisibility of monotonically-increasing integer polynomial

For positive integers $n$ and $x$, let $f_n(x)$ be a polynomial in $x$ of degree $n-1$, such that $f_n(x)$ is monotonically increasing for increasing $x \ge 1$. Now assume that there exist positive ...
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### Suppose $m/l$ is a root of $f$, where $m$ and $l$ are relatively prime integers. Show that $m$ divides $a_0$ and $l$ divides $a_n$.

Suppose $f(x) = a_nx^n + \dots + a_1x + a_0$ is a polynomial with integer coefficients, and suppose $m/l$ is a root of $f$, where $m$ and $l$ are relatively prime integers. Show that $m$ divides $a_0$ ...
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### Digit-sum division check in base-$n$

Several years ago now I realised that for any natural numbers $x$ and $y$ you could write $$x^y=(x-1) \left(\sum_{i=0}^{y-1}x^i\right)+1$$ This shows that $x^y-1$ will always be divisible by $x-1$, ...
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### 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 ...
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### Properties involving prime factorization and divisibility

Can anyone help me out this with proof? Let n be a positive integer greater than 1 with the property that whenever n divides a product ab where a,b ∈ Z, then n divides a or n divides b. Prove ...
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### The Divisors of $s(2s+1)$ and Primes $2n+1$ and $3n+1$ part 1

I want to check my math (and proof) on the following claim. The claim is by way of a computer search and a "hunch". claim: If $s$ is a prime number I write $\varphi_{s} =s(2s+1)$. Let $\tau$ be ...
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### Show that if $d_1e_1=d_2e_2,\ and\ \gcd(e_1,e_2)=1\Rightarrow\ lcm(d_1,d_2)=d_1e_1=d_2e_2$

Show that for positive integers if $d_1e_1=d_2e_2\ and \gcd(e_1,e_2)=1\Rightarrow\ lcm(d_1,d_2)=d_1e_1=d_2e_2$ We know that: $$lcm(d_1,d_2)gcd(d_1,d_2)=d_1d_2$$ but this doesn't help much! What's the ...
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### Find the $\gcd(pq, (p-1)(q-1))$ if $p$ and $q$ are prime. [closed]

Given prime numbers $p$, $q$, how do I prove that $\gcd(pq, (p-1)(q-1)) = p$, $q$ or $1$?
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### probability of divisibility by $5$ [duplicate]

Let $m,n$ be $2$ numbers between $1-100$ . what is the probability that if we select any two random numbers then $5|(7^m+7^n)$ . My attempt last digit should be $5$ or $0$ so $7$ powers follow the ...
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### Doubt regarding divisibility of the expression: $1^{101}+2^{101} \cdot \cdot \cdot +2016^{101}$

In an interesting contest question I recently encountered, I chanced upon a question I couldn't solve. $$\sum^{2016}_{i=1}i^{101}$$ is divisible by: (a)2014 (b)2015 (c)2016 (d)2017 How would I ...
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### Prove that for every positive integer, this polynomial is divisible by 8 [duplicate]

prove that: $$8\mid (n-1)n(n+1)(n+2)$$ I tried to simplify this expression but had no luck.
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### Prove that for every positive integer, this polynomial is divisible by 24. [closed]

Prove that: $$24\mid n^4 + 2n^3 - n^2 - 2n, \quad \forall n\in \mathbb{Z}^+$$ I tried to prove it, but had no luck.
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### Beginner Number Theory Proofs - Common divisors and multiples

I'm taking a mathematics class where we have learned some introductory number theory - but I am having trouble with the whole 'proving this and that' component (most of it lol). Particularly with ...
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### Are there names for any of these four classes of numbers related to divisors and totatives?

Are there names for any of these four classes of numbers related to divisors and totatives? A [insert name here] of $n$ is a positive integer $\leq n$ that isn't a divisor of $n$ and that can be ...
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### Greatest common divisor questions? [closed]

An integer d is a divisor of a ⇔ ____ | ____. Equivalently, d is a divisor of a ⇔ ____ mod ____ = _____. Is it possible for a divisor of a to be bigger than a? The first blank would be d|a, and I am ...
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### One of any consecutive integers is coprime to the rest

After reading this question, I conjectured a generalization of it. Conjecture: Fix $k\in \mathbb N$. Then, for all $n\in \mathbb N$, one of $n+1,\ldots,n+k$ is coprime to the rest. I ...