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

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Efficient way needed

Given N, M find the : GCD(1, 1) * GCD(1, 2) * … * GCD(1, M) * GCD(2, 1) * GCD(2, 2) * … * GCD(2, M) * … * GCD(N, 1) * GCD(N, 2) * … * GCD(N, M) modulo 10^9+7 Constraints: 1 <= N, M <= 2 * ...
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1answer
39 views

How to prove $D(n)<2n(\log\log n)$?

How to prove $D(n)<2n(\log\log{n})$ for all sufficiently large $n$ where $D(n)$ is the Divisor summatory function.
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1answer
68 views

Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$.

Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$. This is a problem from a selection to IMO 2014. I like thinking about this problem, it is ...
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1answer
60 views

product of greatest common divisor

Given two numbers $m$ and $n$ how can we calculate the gcd product of any two numbers i.e, $\operatorname{gcd p}(n,m)=\gcd(n,1)\gcd(n,2)\cdots\gcd(n,m)$ where gcd is the greatest common divisor? Can ...
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55 views

Prove the following : [duplicate]

Prove the following : $$ {{n}\choose{7}}-\left \lfloor{\frac{n}{7}}\right \rfloor $$ is divisible by 7.
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55 views

Find all integers $n$ (positive, negative, and zero) so that $n^2 + 1$ is divisible by $n + 1$.

I found $n=0, n=1, n=-2,$ and $n=-3$, but I am having trouble showing that these are the only four. I was thinking about maybe showing that no integer on the intervals $(-\infty, -3), (-3, -2), (-2, ...
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1answer
85 views

How many regulars do the primorials 223092870 and 6469693230 have?

Regulars = Divisors + Semidivisors http://global.britannica.com/EBchecked/topic/496213/regular-number So for example: 6 has 5 regulars: 1, 2, 3, 4, 6. 8 has 4 regulars: 1, 2, 4, 8. 9 has 3 ...
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68 views

Among the superior highly composite numbers, which are the most divisor dense numbers?

I’m searching for the most divisor dense natural numbers. Firstly we have the highly composite numbers: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, … But ...
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3answers
71 views

Is x/x equal to 1

My question is whether $x/x$ is always equal to 1. I am mostly intersted in real numbers and particularly wonder whether $x/x$ is defined at $x=0$. On one hand the division should simplify to 1, on ...
2
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2answers
117 views

Asymptotic divisor function / primorials

Let $p_n\#\equiv\prod_{k=1}^{n}p_k$ (primorial), and $\sigma(n)=\sum_{d|n}^{}d$ (divisor function). Could someone please tell me what the general asymptotic of $\dfrac{\sigma(p_n\#)}{p_n\#}$ is? It ...
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1answer
28 views

For how many values of $a,b,c\in(1,2\ldots,p-1)$ does $p$ | $({a^2}-bc)$ where $p$ is an odd prime number

In a mock test for an entrance exam I am preparing for came the following question: Let $p$ be an odd prime number and $T_p$ be the following set of matrices $$ T_p= \left( ...
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2answers
26 views

Divisibility Problem: How can I solve this?

Suppose that $a,b,q,r$ are any integers such that $b > 0$ and $a = bq + r$, with $0\le r<b$, and suppose $b|a$. Must it be the case that $r = 0$? Justify your answer. Can anyone please let me ...
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33 views

The elegant expression in terms of gcd and lcm - algebra - (2)

Definition: suppose a quantity $P$ is identified by $$ \frac{P}{k}\simeq \frac{P}{k}+1 $$ what we mean is that $$ P= 0\pmod{k}. $$ That means that when $P \to P+k$, then $$ \frac{P}{k}\to ...
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1answer
44 views

Verify my proof on elementary number theory

I've tried to prove this theorem, which is very simple, but is a kind of practice for me. Let $a,b$ be two positive integers. Therefore, if $a+b$ is a composite number, $frac(\frac{a}{l}) + ...
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54 views

Vieta jumping with non-monic polynomials

I have recently discovered Vieta jumping as a problem-solving technique. In order to teach myself about it, I have located most (all of?) the standard references, both here on MSE and "out there" (via ...
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36 views

Moving up the Y axis the lengh of the hypotenuse of a right triangle

If i have a right triangle ABC with B being the right triangle and length AB = 50 and length BC = 50. Based on the Cartesian coordinate system if i wanted to move up the Y axis the length of the ...
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1answer
39 views

The elegant expression in terms of gcd and lcm - algebra

Given three positive integer numbers $k_1$, $k_2$, $k_3$, we may denote their greatest common divisor(gcd) by $\gcd(k_i,k_j)\equiv k_{ij}$ for gcd of a two pair of number $k_i,k_j$. ...
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1answer
26 views

$(x+b)^3\mid P(x)+a$ and $(x-a)^3\mid P(x)-a$

$a,b\in\mathbb{C}$, $b!=0$ I need to find all the polynomials $P$ of degree $5$ verifying: $ \begin{cases}(x+b)^3\mid P(x)+a\\ (x-b)^3\mid P(x)-a\end{cases} $ PS : there was en error, i fixed it ...
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2answers
86 views

How does $n!^2$ divide $(2n)!$? [duplicate]

How can I show that $(n!)^2$ divides $(2n)!$, where $n$ is a natural number? So far I've noticed that we can rewrite $\dfrac{(2n)!}{(n)!^2}$ as a combination and we know that combinations are always ...
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1answer
44 views

Properties of Integers

A theorem presented in my discrete math book. Let $d$ be the smallest positive integer of the form $ax + by$. Then $d = \gcd(a,b)$, where gcd means greatest common divisor. I don't understand ...
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Do Question's Given GCD Statements Imply these New GCD Statements?

Are the following statements true or false, where $a$ and $b$ are positive integers and $p$ is prime? In each case, give a proof or a counterexample: (b) If $\gcd(a,p^2)=p$ and ...
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1answer
59 views

How to show $(n-1)^3n^3(n+1)^3$ is divisible by 7 and 9?

Yeah it looks like a basic, really elementary question, but i'm having hard time with it. First i tried to show that it's divisible by 9 $$(n-1)^3n^3(n+1)^3 = ((n+1)(n-1))^3n^3 = (n^2-1)^3n^3 = ...
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1answer
32 views

Number Theory Divisibility Question

(From Math Challenge II Number Theory packet) Given that $a,b,n$ are positive integers. Assume that for any positive integer $k\neq b, (k-b)\mid(k^n-a)$, the which of the following must be true? ...
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WordProblem on factors and remainder theorem

Mr.Chaalu while travelling by Ferry queen has travelled the distance one Kilometer more, than the fare he paid per km. Initially he had total amount of Rs.350/- in his wallet. Now he is only left with ...
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1answer
57 views

Prove that $n$ is divisible by $6$

Problem: Let $x^2+mx+n$ and $x^2+mx-n$ give integer roots where $(m,n)$ are integers. Show that $n$ is divisible by $6$ My attempt: Since the roots are integers then the discriminants of both the ...
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2answers
54 views

Divisibility of sum of powers: $\ 323\mid 20^n+16^n-3^n-1\ $ for which $n?$

I found this question in my Math Challenge II Number Theory packet: Find all positive integers $n$ that satisfy $323|20^n+16^n-3^n-1$. I don't even have any idea how to approach this question. Any ...
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63 views

How to prove that at least one of $a,b,c,d$ is not divisible by $ad-bc$ if $ad-bc>1$?

we have $ad-bc >1$ is it true that at least one of $a,b,c,d$ is not divisible by $ad-bc$ ? Thanks in advance. Example: $a=2$ , $b = 1$, $c = 2$, $d = 2$, $ad-bc = 2$ so $b$ is not divisible by ...
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4answers
96 views

Show that $(x + 1)^{2n + 1} + x^{n + 2}$ can be divided by x^2 + x + 1 without remainder

I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as P = Q*L + R I am unable to ...
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5answers
347 views

How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$?

For my proof I distinguished the two possible cases which derive from $7 \mid a^2+b^2$: Case one: $7\mid a^2$ and $7 \mid b^2$ Case two (which (I think) is not possible): $7$ does not divide $a^2$ ...
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2answers
51 views

If a natural number $x$ is divisible by $3$

Is the sentence If a natural number $x$ is divisible by $3$ then, if it is not divisible by $3$ then it is divisible by $5$ true or false?
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1answer
65 views

Factors of a perfect square plus one

For large integer $a$, small integer $d$, consider the following quantity: $$a^2+d$$ What are the best lower bounds one can get for the sum $l+m$, where integers $l,m$ are such that: $$lm=a^2+d$$ ...
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105 views

How to solve the equation $n^2 \equiv 0 \pmod{584}$?

Well, I've confused when trying to solve this equation can anybody help me : $n^2 \equiv 0 \pmod{584}$ I tried to factorize the $584$ i got $584=2^3\times73$. so $n^2$ has to be divisible by $2^3$ ...
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2answers
36 views

Remainder question with $6!$ and 7

Find the remainder when $6!$ is divided by 7. I know that you can answer this question by computing $6! = 720$ and then using short division, but is there a way to find the remainder without using ...
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0answers
26 views

Using $ \gcd(a,b) = \gcd(b,r) $ if $ a \equiv r \pmod b$ for GCD?

It should be true that $\gcd(a,b) = \gcd(b,r) $ if $ a \equiv r \pmod b$. But: How can I use this equality to compute the GCD of $a$ and $b$? It seems as if $r$ is of the form $r = k\cdot b + s$ ...
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Divisibility question

I didn't understand how they reached at this conclusion: If $b\mid x-y$, $b\in \mathbb{N}$, $b\geq 2$ In this inequation: $$-(b-1)\leq x-y\leq b-1$$ The only integer divisible by $b$ is zero. (Why ...
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1answer
65 views

Pythagorean quadruple generators with a gcd relation

For non-negative integers $m,n,q,p$ with $\gcd(m,n,q,p)=1$, assume we have: $$\gcd(mq+np,b)=|nq-mp|$$ for some integer $$b<mq+np$$ and that $$8\nmid\,mq+np,$$ $$m+n+p+q\equiv 1\mod 2.$$ Can ...
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115 views

To find relatively prime ordered pairs of positive integers $(a,b)$ such that $ \dfrac ab +\dfrac {14b}{9a}$ is an integer

How many ordered pairs $(a,b)$ of positive integers are there such that g.c.d.$(a,b)=1$ , and $ \dfrac ab +\dfrac {14b}{9a}$ is an integer ?
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Why define $\gcd(r,s)$ as a positive generator $d$ of the cyclic group $H=\{nr+ms|n,m\in\mathbb{Z}\}$?

This is in regards to definition 6.8, p. 62 from Fraleigh's "A first course in abstract algebra". 6.8 Definition Let $r$ and $s$ be two positive integers. The positive generator $d$ of the ...
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52 views

Greatest Common Divisor Divisibility Question

Can we characterize the pairs of positive integers $b<a$ such that: $b|(a^2+\gcd(a,b))$
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1answer
216 views

Curious number theory problem

$k,m,n\in\mathbb{N}$ satisfy $k^{m+n}=nm^n$. How can I show that $m=k$ and $n=k^k\,?$
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3answers
38 views

Prove that if m is prime and m|kl then either m|k or m|l.

Proofs homework question, here's what I've figured out thus far. Suppose m doesn't divide k. We need to then prove that m|l. If m doesn't divide k and m is a prime then we know m and k are co-prime ...
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2answers
590 views

Does $n \mid 2^{2^n+1}+1$ imply $n \mid 2^{2^{2^n+1}+1}+1$?

There are two ways to try to prove this. One is in the title, the other is its de Morgan counterpart: $n \nmid 2^{2^{2^n+1}+1}+1 \implies n \nmid 2^{2^n+1}+1$. Disproving it requires only one example ...
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How to prove that if a number is divisible by two other numbers, then it is divisible by there product

I would like to prove if $a \mid n$ and $b \mid n$ then $a \cdot b \mid n$ for $\forall n \ge a \cdot b$ where $a, b, n \in \mathbb{Z}$ I'm stuck. $n = a \cdot k_1$ $n = b \cdot k_2$ $\therefore a ...
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GCD of this polynomial

So here is the exact question that i am having trouble on: "Extend the Euclidean algorithm to polynomials and find the greatest common divisor of: $3x^5-10x^4-4x^3-14x^2-7x-4$ and ...
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2answers
57 views

Solving a problem with a diophantine equation without trial and error.

I have the following problem: A teacher bought toys for the students of an academy, every toy for a boys costs $290$ and every toy for a girl costs $330$. If he spends $24300$, how many of each ...
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3answers
75 views

Suppose that $2^b-1|2^a+1$. Show that $b = 1$ or $2$.

I'm stuck with this one. I would appreaciate any idea how to prove this.
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3answers
122 views

How prove this $\binom{n}{m}\equiv 0\pmod p$

let $p$ is prime number,and such $p\mid n,p\nmid m,n\ge m$ show that $$p\>\Big|\>\binom{n}{m}$$ I know that: if $p$ is prime number,then $$\binom{n}{p}\equiv \left[\dfrac{n}{p}\right] \pmod ...
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3answers
41 views

How to find exponent of a number in a combination?

How do I find the exponent of $7$ in $^{100}C_{50}$ that is, $\dfrac{100!}{(100-50)!\cdot 50!} =\dfrac{100!}{50!\cdot 50!}$, this question was out of the blue, and I haven't been able to find any ...
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1answer
68 views

Prove something is divisible by a prime

Let $p$ be a prime. Prove that $\sum_{k=j}^{p-1} \frac{k!}{ j! (k-j)! }$ is divisible by $p$ $\forall$ $j \in \{0, ..., p-2\}$. Where this problem comes from: I am trying to prove that ...
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6answers
95 views

If $n$ is a perfect square and a perfect cube then $7$ divides $n(n-1) $

If a positive integer $n$ is both a perfect square and a perfect cube , then is it true that $7$ divides $n(n-1)$ ?