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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1answer
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Find the smallest $n$ such that the $n$-th prime $p_n \equiv 330 \mod n $.
Find the smallest $n > 1$ such that the $n$-th prime $p_n \equiv 330 \mod n $.
I was investigating the remainders when the $n$-th prime is divided by $n$. For every positive integer $a < 330$, ...
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1answer
24 views
Divisor Properties
This is a general question regarding divisibility
If $c|a$ and $c|b$, does $c|a+b$?
Where $a,b,c$ are integers
If yes, does this also hold for general rings?
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1answer
38 views
Perfect squares and divisor
Let $n$ be a positive integer and let $d$ be a positive divisor of $2n^2$. Prove that $n^2+d$ is not a perfect square.
My working:
$d \mid 2n^2$
Let $d \cdot k=2n^2 \implies d=\dfrac {2n^2}k$
...
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4answers
45 views
I am looking for a proof of a certain set being divisible by 7
I am looking for a proof of this statement:
$$7\mid{3^{6k+2}-{2^{6k+1}}}$$
By trial and error I can see that it holds but I cant figure out anyway to prove it or cant seem to be able understand why.
...
2
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1answer
69 views
Divisibility of $x^2+y^2$ by prime $p$
I've read the following fact on my number theory textbook, there's no proof on the book of such result, I tried working it out on my own but I'm kinda lost, the lemma is the following:
Given two ...
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0answers
10 views
How can the facial thirds be measured without reference points? [closed]
So I have an interesting question regarding facial thirds. My concern is about how a quantifiable base of the sum of all three thirds divided can be reliably accurate insofar as to properly measure ...
3
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0answers
88 views
Elementary number theory proofs
I am a freshman studying computer science, and I am supposed to solve this problem for my '(introduction to) elementary number theory' course. Could someone give me a hint or two on how to solve the ...
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1answer
40 views
Does there exist a composite, deficient, odd number that is divisible by the sum of its proper factors?
Consider $n \in \mathbb{N}$.
Define the aliquot sum function $s(n)$ to be the sum of the proper divisors of $n$ (the divisors not including $n$ itself). Call $n$ deficient if $s(n) < n$, abundant ...
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1answer
65 views
Two friends have $2$ natural written on their forehead. One is $2$ times the other + $1$. They can raise their hands.
The problem:
Two friends have $2$ natural written on their forehead. One of them is $2$ times the other + $1$. Let's call them $X$ and $2X + 1$. They have to come up with a strategy to guess their ...
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2answers
111 views
Show that $(\binom{p^2}{p} -p ) $ is divisible by $p^5$, for every prime number $p, p\ge 5$
Show that $(\binom{p^2}{p} -p ) $ is divisible by $p^5$, for every prime number $p, p\ge 5$.
I have a combinatorics problem, and this is what it reduces to. I am not quite sure how to link the fifth ...
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2answers
33 views
Divisibility Theory [closed]
The question asks me to prove that no integer of the $8k+5$ where k is a positive integer, can ever be a perfect square.
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1answer
7 views
Return the number of integers within the range of a and b that are divisible by x
So I have a question here:
Return the number of integers within the range of a and b that are divisible by x.
So I have, a = 0, b = 17 and x = 17.
Apparently the answer is 2.
I understand that 17 /...
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1answer
53 views
Number of divisors of $ 20^{20} $ with exactly $20$ divisors
How many positive integers $x$ with $x\mid 20^{20}$ have exactly $20$ divisors ?
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4answers
183 views
A simple doubt in number theory problem.
I considered an even number $n\geq 10$, where it is divisible by some positive integer $k$.
Also, $k$ divides $\frac{n}{2}$. Then $n = kq\implies k\cdot\frac{q}{2} = \frac{n}{2}$.
Can we say that $q$...
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0answers
58 views
If $y^2-x^2\bigm|2^ky-1$ and $2^k-1\bigm|y-1$ then $y=2^k$ and $x=1$
Suppose that $k\geq2$ and $0<x<y$ and $y^2-x^2\bigm|2^ky-1$ and $2^k-1\bigm|y-1$. Is it necessarily the case that $x=1$ and $y=2^k$?
I've tested this up to $k\leq50$ and $y\leq10000$ but I ...
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4answers
77 views
Find all pairs $(m, n)$ of positive integers such that $m$ divides $8n+1$ and $n$ divides $8m+1$
I've found the pairs $(1,3),(1,9),(3,25)$ and $(13,21)$ up to order. But I have no idea how to prove that there are not other solutions. Any hints...? I've been trying for a few days but all I came up ...
3
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1answer
124 views
Are are infinitely many intimate pairs of integers?
Using the standard statistical definitions, the variance of $x_1, x_2, \ldots, x_n$ and the squared errors about its mean $\mu$ are given by $\sigma^2 = \sum_i(x_i - \mu)^2/n$ and $\delta_2 = \sum_i(...
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1answer
26 views
Determine a property of $S_b(n)$, which is the sum of the digits of $n$ when $n$ is expressed in base $b$
The original question Let $S_b(n)$ be the sum of the digits of $n$ when $n$ is expressed in base $b$. was asked by Anson Chan on Jan. 24, 2019. Since it was deemed to not have enough context, it was ...
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2answers
81 views
Proof for divisibility of polynomials. [closed]
I have no idea how to proceed with the following question. Please help!
"Prove that for any polynomial $ P(x) $ with real coefficients, other than polynomial $x$, the polynomial $ P(P(P(x))) − x $ is ...
2
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3answers
41 views
Prove $a^n+1$ is divisible by $a + 1$ if $n$ is odd [duplicate]
Prove $a^n+1$ is divisible by $a + 1$ if $n$ is odd:
We know $a$ cannot be $-1$ and the $n \in \mathbb{N}$.
Since $n$ must be odd, we can rewrite $n$ as $2k+1$. Now we assume it holds for prove that ...
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0answers
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A mysterious fact related to digital root of a 'special incorrect division' and 'correct division'.Whats the reason behind it?
I observed that the digital root of an 'incorrect division' obtained by dividing a number in a 'special way'/'particular manner' and the digital root of the correct division are always same.
The ...
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3answers
87 views
$8^n-3^n$ Divisible by 5 - Proof Verification.
Statement: $\frac{8^k-3^k}{5}=M, M\in\mathbb{N}$
Base case: $P(1): \frac{8-3}{5}=1\in\mathbb{N}$
Assume $P(n): \frac{8^n-3^n}{5}=N$
Then, $P(n+1)=8^{n+1}-3^{n+1}=5K$, where $K$ is in terms of $M$
...
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1answer
21 views
Given a number $N$ and two numbers lower than $N$, what are all of the combinations of those numbers that sum to $N$?
I tried looking around the website for a while but I was unable to find anything that matched this problems specifically. So if someone here knows of a question that is the same as mine, please link ...
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3answers
242 views
Can divisibility rules for digits be generalized to sum of digits
Suppose that we are given a two digit number $AB$, where $A$ and $B$ represents the digits, i.e 21 would be A=2 , B=1. I wish to prove that the sum of $AB$ and $BA$ is always divisible by $11$.
My ...
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1answer
33 views
Prove that if x is odd and y is even, then gcd(x+y,x-y)=gcd(x,y)
It is trivial to prove that gcd(x,y) divides gcd(x+y,x-y). How is it possible to prove gcd(x+y,x-y) divides gcd(x,y)? I don´t know how to use the fact that x is odd and y is even. Can anybody help me ...
2
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1answer
34 views
If $a \mid c$ and $b \mid c$ where $a, b, c \in \mathbb{N}$, under what conditions does it follow that $a \mid b$?
The following question is pretty basic, and the underlying idea was used in the "proof" of a statement in this hyperlinked answer to another MSE question.
The question is as follows:
If $a \mid c$ ...
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6answers
102 views
$31$ divides to $28!+233$? [duplicate]
How to prove that $31$ divides to $28!+233$?
I have tried to use Wilson's theorem and the primes decomposition theorem but I have not had success.
Thanks for your help
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2answers
32 views
Does division by zero imply a different type of division in this situation
Firstly, I would like to apologize if this is somehow addressed in one of the many many explanations about why division by 0 is impossible that appear on this site. I have not yet found one that ...
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2answers
45 views
Number theory divisibility check question
N = $2^{744} - 1$.
Prove N is divisible by $2^{93}+2^{47}+1$. I have no idea how to proceed. (edit: removed first part as I got the answer)
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1answer
31 views
Is there a name for the largest factor $f$ of a number $n$ so that $n/f \ge f$?
TL;DR: Is there a name for the largest value of $f$ for which $f|n$ and $n/f \ge f$? Or a name for the smallest value of $f$ for which $f|n$ and $n/f < f$?
Additional clarification. This question ...
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0answers
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On two nested radicals and divisibility
The last days I was playing around with two nested radicals which, as I learned here, can be simplified:
$$u(x) =\sqrt{x + \sqrt{x +\sqrt{x +\sqrt{x +...}}}} = \frac{1}{2}(1+\sqrt{1+4x})$$
$$l(x) = \...
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1answer
60 views
Find all triplets $(a,b,c)$ less than or equal to 50 such that $a + b +c$ be divisible by $a$ and $b$ and $c$.
Find all triplets $(a,b,c)$ less than or equal to 50 such that $a + b +c$ be divisible by $a$ and $b$ and $c$.(i.e $a|a+b+c~~,~~b|a+b+c~~,~~c|a+b+c$) for example $(10,20,30)$ is a good triplet. ($10|...
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3answers
95 views
Find the remainder when the polynomial $1+x^2+x^4+x^6+…+x^{22}$ is divided by $1+x+x^2+x^3+…+x^{11}$
Find the remainder when the polynomial $$1+x^2+x^4+x^6+....+x^{22}$$ is divided by $$1+x+x^2+x^3+...+x^{11}$$
$1+x^2+x^4+x^6+....+x^{22}=\frac{x^{24}-1}{x^2-1}$
$1+x+x^2+x^3+...+x^{11}=\frac{x^{12}-...
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2answers
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Prove that if $f(x)$ divides $g(x)$ and $f(x)$ divides $h(x)$, then $f(x)$ divides $s(x)g(x) +t(x)h(x)$.
Let $F$ be a field. Prove that for all polynomials $f(x), g(x), h(x) \in {F}[x]$, if $f(x)$ divides $g(x)$ and $f(x)$ divides $h(x)$, then for all polynomials $s(x), t(x)\in {F}[x]$, $f(x)$ divides $s(...
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0answers
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Non-trivial divisors
I want to find out the number of integers whose biggest non-trivial divisor is exactly $k$ times the smallest non-trivial divisor of that integer.
My thoughts are, that the smallest divisor $n$ has ...
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3answers
47 views
How to eliminate prime factors from algebraic integers?
I'm trying to eliminate prime factors from algebraic integers.
Is the following true without further restrictions? And how can I prove it?
Let $p,N,M\in\mathbb Z$, let $p$ be prime such that $gcd(p;N)...
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0answers
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Nice tuples! A number theory problem. [duplicate]
How many tuples $(a,b,c)$ could be found so that $a+b+c$ is divisible by $a$ and $b$ and $c$.
$a, b, c\; $ are natural numbers less than or equal to $50$.
I'm stuck on this homework problem. ...
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1answer
45 views
Divisibility of fourths by seven
I am otherwise very good in mathematics, but recently I came upon a problem that I just can't solve. Do you have any idea how to solve it?
If $a^2 + b^2 + c^2$ is divisible by 7, prove that $a^4 + b^...
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2answers
108 views
Prove that $\frac{(72!)}{(36!)^2}-1$ is divisible by 73 [closed]
Prove that $\frac{(72!)}{(36!)^2}-1$ is divisible by 73.
My approach is as follow $73n=\frac{(72!)}{(36!)^2}-1$ I tried remainder theorem but could not prove it.
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2answers
27 views
Prove that $M_n\mid (i+n)(i-n)$ and $M_n\mid(i+n-1)(i-n+1)$
My friend gave me a problem:
Prove that if $M_n$ is the $n^\text{th}$ odd number, then for all integers $i$, $$M_n\mid (i+n)(i-n)\quad\text{and}\quad M_n\mid(i+n-1)(i-n+1)\tag*{$[1]$}$$
The ...
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1answer
59 views
How l can draw Hasse diagram
How can l draw a Hasse diagram of the divisibility relation, when $$B=\{2,4,5,6,7,10,18,20,24,25\}$$
Would any help, thank you.
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2answers
54 views
How to determine the smallest value of $N=n^4+6n^3+11n^2+6n$ if 13 and 19 both divide N?
I tried to solve for an integer solution by making N equal to multiples of 247 but this is not leading me anywhere. I then tried using the tests for divisibility which did not seem to lead me anywhere ...
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3answers
103 views
Hopeless Numbers
Beatriz Viterbo has called a positive integer which is not divisible by any of the ($2^n$, where $n$ is the number of its digits) numbers that result by introducing a plus or minus sign to the left of ...
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1answer
15 views
distribute items into containers evenly without splitting
Im trying to figure out how to place a number of items into containers evenly but without splitting.
Easy example:
11 items, 2 containers
11/2 = 5.5
so even distribution would look like this (...
4
votes
3answers
154 views
Isn't there any divisor $k$ of $n^4$ such that $n^2-n<k<n^2$?
I did some experiment with my Python script to find a number which could divide $n^4$ in this interval ($n^2-n$, $n^2$).
I watched the form of prim factors of the numbers in this ($n^2-n$, $n^2$) ...
1
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7answers
82 views
How to find divisibility of a very large number
I was asked this question in a test:
The number $111111...111$ ($1$ comes 91 times) is a:
A) Prime number
B) Composite Number
C) divisible by $\frac{10^7 -1}{9}$
The answer is B ...
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1answer
66 views
Prove that for every positive integer $n>0$, $3\sum\limits_{i=1}^n i^5 $ is divisible by $\sum\limits_{i=1}^n i^3$ [closed]
Can you help me with this problem :
Prove that for every positive integer $n>0$,
$3\sum\limits_{i=1}^n i^5 $ is divisible by $\sum\limits_{i=1}^n i^3 $
1
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2answers
149 views
Describe the Natural density of $p$ which divides natural numbers of the form $n^2+1$?
We want to find the numbers that divide natural numbers in the form of $n^2+1$ and solve for their natural density.
Using Wolfram Mathematica, I found divisors from $n=0$ to $1000000$ and eliminated ...
1
vote
2answers
55 views
Prove that if there are integers $m$ and $n$ such that $am +bn =1$ then $a$ and $b$ are coprime.
Suppose $a,b \in \mathbb{N}$. Prove that if there are integers $m$ and $n$ such that $am +bn =1$ then $a$ and $b$ are coprime.
I came up with the following proof, but I am sure a shorter argument ...
1
vote
1answer
40 views
How to prove that if $(ab,n)=1$ then, $(r,n)=1$? [duplicate]
Let $ab=nq+r$ where all variables represent integers with $0\leq r<n$. If $(ab,n)=1$ then how to prove that $(r,n)=1$? I need to prove this to help me understand the proof of Euler's theorem better....
