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

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2
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5answers
132 views

Prove that if $6 \mid (2a + 4)$ and $9 \mid (12 + 3b)$ then $3 \mid (a + b)$

Could you help me with the problem below? Prove that if $6 \mid (2a + 4)$ and $9 \mid (12 + 3b)$ then $3 \mid (a + b)$. Thank you!
1
vote
2answers
111 views

Let $d$ be a positive odd integer. Prove that there exists a positive $n \le d$ such that $d | 2^n − 1$.

I'm stuck on this question from my textbook which doesn't even have a solution. Any ideas ?. Help would be much appreciated. $$ d\, \left\vert\right.\, \left(2^{n} − 1\right) $$
1
vote
3answers
120 views

Basic Modulo Question

I've been having trouble with this example while studying for my exams. Why is $$2023^{2297}\equiv 20 \pmod{3953}\;?$$ Thanks so much for any help I can get! The examples solves the answer by ...
0
votes
4answers
109 views

Prove that if a and b are integers, then there are unique integers q and r such that $a = bq + r$, $-|b|/2 < r \le |b|/2$ [closed]

Prove that if a and b are integers, then there are unique integers q and r such that $$a = bq + r,$$ with the restriction that$$-|b|/2 < r \le |b|/2$$
0
votes
1answer
90 views

Two issues of Number Theory

Knowing in Fibonacci sequence$$u_n\mid u_m\Longleftrightarrow n\mid m$$ Question 1: In Fibonacci sequence, show that $$5\mid u_m\Longleftrightarrow 5\mid m$$ Show: $\Longrightarrow$ In ...
0
votes
1answer
76 views

Prime divisibility in a prime square bandtwidth

I am seeking your support for proving (or fail) formally the following homework: Let $p_j\in\Bbb P$ a prime, then any $q\in\Bbb N$ within the interval $p_j<q<p_j^2$ is prime, if and only if ...
33
votes
4answers
1k views

How does the divisibility graphs work?

I came across this graphic method for checking divisibility by $7$. $\hskip1.5in$ Write down a number $n$. Start at the small white node at the bottom of the graph. For each digit $d$ in ...
21
votes
10answers
1k views

Prove if $56x = 65y$ then $x + y$ is divisible by $11$

If $x$ and $y$ are natural numbers, and $56x = 65y$, prove that $x + y$ is divisible by $11$. I tried taking the $\gcd(56x,65y)$ using the Euclidean algorithm, but I got nowhere with it and do not ...
4
votes
1answer
218 views

Divisibility of $2^n - 1$ by $2^{m+n} - 3^m$.

For what values of $m,n$ natural, do $2^n - 1$ is divisible by $2^{m+n} - 3^m$? Thank you very much.
9
votes
3answers
218 views

Show that $(n!)^{(n-1)!}$ divides $(n!)!$

Show that $(n!)^{(n-1)!}$ divides $(n!)!$ I found this question in a text he was reading about BREAKDOWN OF PRIME FACTORS IN FACT, I decided many years, have posted some here to help, but this ...
8
votes
2answers
2k views

Divisibility Rules for Bases other than $10$

I still remember the feeling, when I learned that a number is divisible by $3$, if the digit sum is divisible by $3$. The general way to get these rules for the regular decimal system is ...
6
votes
1answer
76 views

Bezouts Identity for prime powers

I have two prime powers $2^n$ and $5^n$ for some arbitrary $n$. Their gcd is $1$ but how do I get their integer linear combination which is $1$ in terms of $n$. In other words what will be the ...
3
votes
6answers
1k views

Trick to find multiples mentally

We all know how to recognize numbers that are multiple of $2, 3, 4, 5$ (and other). Some other divisors are a bit more difficult to spot. I am thinking about $7$. A few months ago, I heard a simple ...
17
votes
3answers
661 views

My first induction proof (very simple number theory). Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
6
votes
3answers
373 views

If $R$ is a commutative ring with identity, and $a, b$ are its elements that are divisible by each other, is it true that they must be associates?

Thank you very much! My problem is: If $R$ is a commutative ring with identity, and $a, b$ are its elements that are divisible by each other, is it true that they must be associates? Here, $a$ ...
3
votes
2answers
82 views

$\gcd(c^a + 1, c^b + 1)$ for even $a$ and $b$?

Following on this question, what is the Greatest Common Denominator of $c^a + 1$ and $c^b + 1$, where $a, b, c \in N$. I know that for odd a and b, we have $\gcd(c^a + 1, c^b + 1) = c^{\gcd(a, b)} + ...
3
votes
1answer
81 views

Prove that $x^4+x^3+x^2+x+1 \mid x^{4n}+x^{3n}+x^{2n}+x^n+1$

Problem: Prove that $x^4+x^3+x^2+x+1$ divides $x^{4n}+x^{3n}+x^{2n}+x^n+1$ for all positive $n$ that are not multiples of $5$. I'd like to get some pointers about how to solve this. No full ...
2
votes
5answers
159 views

law of divisibility on $37$

how to find and prove law of divisibility on $37$? Thanks in advance. Added:---- how to prove for$37$ that: Split off the last digit, multiply by 11, and subtract the product from the number that is ...
2
votes
2answers
368 views

If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$

How to prove that: If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$ This statement is generalization of the statement from my previous question. I have checked for many $(a,b)$ ...
1
vote
5answers
63 views

Prove if $a\mid b$ and $b\mid a$, then $|a|=|b|$ , $a, b$ are integers.

Form the assumption, we can say $b=ak$ ,$k$ integer, $a=bm$, $m$ integer. Intuitively, this conjecture makes sense. But I can't make further step.
14
votes
6answers
848 views

Divisibility criteria of 24. Why is this?

I am currently familiar with the method of checking if a number is divisible by $2, 3, 4, 5, 6, 8, 9, 10, 11$. While Checking for divisibility for $24$ (online). I found out that the number has to ...
7
votes
7answers
1k views

Proof for divisibility by $7$

One very classic story about divisibility is something like this. A number is divisible by $2^n$ if the last $n$-digit of the number is divisible by $2^n$. A number is divisible by 3 (resp., by ...
4
votes
1answer
97 views

how to find all $n \in \Bbb N$ such that $n(n+1)\mid(n-1)! $

how to find all $n \in \Bbb N$ such that $n(n+1)\mid(n-1)! $
3
votes
2answers
683 views

Trying to prove that $p$ prime divides $\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1$

So I'm trying to prove that for any natural number $1\leq k<p$, that $p$ prime divides: $$\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1$$ Writing these choice functions in ...
2
votes
3answers
78 views

For every integer $n$, the remainder when $n^4$ is divided by $8$ is either $0$ or $1$.

I am trying to prove the following statement: For every integer $n$, the remainder when $n^4$ is divided by $8$ is either $0$ or $1$. So far I have figured out that $n^4 = 8m$ or $n^4 = 8m + ...
2
votes
2answers
246 views

Determine the number of factors for extremely large numbers.

An offshoot from a related question, is there a way to determine the number of possible factors (odd, even, prime, etc.) for extremely large integers without actually factoring them? Even an ...
2
votes
3answers
313 views

Factoring extremely large integers.

The question is about factoring extremely large integers but you can have a look at this question to see the context if it helps. Please note that I am not very familiar with mathematical notation so ...
2
votes
3answers
532 views

16 digit numbers divisible by 17

I wanted to know about the $16$ digit numbers those are divisible by $17$ and when this $16$ digit number is broken in groups of $4$ those groups of four are also divisible by $17$ and a check to ...
1
vote
2answers
74 views

Extended Euclidean Algorithm problem

I'm confused about how to do the extended algorithm. For example, here's the gcd part gcd(8000,7001) $$\begin{align}8000 &= 7001\cdot1 + 999\\ 7001&=999\cdot 7+8\\ 999&=8\cdot 124+7\\ ...
1
vote
2answers
268 views

Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n [duplicate]

Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n Hi everyone, for the proof to the above question, Can I assume that since $(a, b) = ...
1
vote
2answers
179 views

Rabin's test for polynomial irreducibility over $\mathbb{F}_2$

I know that $f(x) = x^{169}+x^2+1$ is a reducible polynomial over $\mathbb{F}_2$. I want to show this using Rabin's irreduciblity test. First, I then have to check if $f$ is a divisor of ...
0
votes
2answers
77 views

Show that if $\gcd(a,b)=1$ and $p$ is an odd prime, then [duplicate]

Show that if $\gcd(a,b)=1$ and $p$ is an odd prime, then ${\gcd(a+b,}\frac{a^p +b^p}{a+b}$$) = 1$ or $p$ Sorry about the duplicate In another answer, however, the sum $\sum\limits_{k=0}^{n-1} ...
0
votes
1answer
46 views

Greatest common divisor of $3$ numbers

Let $a,b, c$ belong to $\mathbb Z$ such that $(a,b,c) \neq (0,0,0)$. Define the [highest common factor] greatest common divisor ${\rm gcd}(a, b, c)$ to be the largest positive integer that divides $a, ...
0
votes
1answer
45 views

Inductive proof for gcd

Proof that $(\forall n \in \Bbb N)[\gcd(n,(25n+1)^3)=1]$ By the inductive method: $p(n):\gcd(n,(25n+1)^3)=1$ $p(1): \gcd(1,26^3)=1 \implies p(n)\equiv \text{True}$ $p(n): $I assume that $p(n)\equiv ...
0
votes
1answer
52 views

What is the concept behind divisibility of large numbers that contain only the digit 1?

An example question I found in a text book is : The 300 digit number with all digits equal to 1 is : A) Divisible by neither 37 nor 101 B) divisible by 37 but not by 101 C) divisible by 101 but not ...
0
votes
4answers
322 views

Divide by a number without dividing.

Can anyone come up with a way to divide any given x by any given y without actually dividing? For example to add any given x to any given y without adding you would just do: $x-(-y)$ And to ...
-2
votes
6answers
195 views

What is the division of $1/0$? [duplicate]

It's approximate value, its infinite I know it but I want to know atleast the value upto $7$ decimal values.
6
votes
4answers
788 views

If $n = m^3 - m$ for some integer $m$, then $n$ is a multiple of $6$

I am trying to teach myself mathematics (I have no access to a teacher), but I am not getting very far. I am just working through the exercises at the end of the book's chapter, but unfortunately ...
5
votes
2answers
88 views

gcd Calculations

Let $a, b, c$ be integers. Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$ First time asking here. I'm not sure what your policies are on general homework help but I truly am ...
4
votes
6answers
194 views

Proof that $a^5 b - b^5 a$ is divisible by $30$ for any integers $a$ and $b$

I am trying to prove that $a^5\times b - b^5\times a$ is divisible by $3$. The actual task is to prove divisibility by $30$ but I have managed to prove that the expression is divisible by $5$ and $2$. ...
4
votes
2answers
118 views

Proving $\gcd( m,n)$=1 [duplicate]

If $a$ and $b$ are co prime and $n$ is a prime, show that: $\frac{a^n+b^n}{a+b}$ and $a+b$ have no common factor unless $a+b$ is a multiple of $n$ Also enlighten me why $n$ has to be prime so that ...
4
votes
3answers
273 views

Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]

The problem is following, prove that: $$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$ I've tried solving this problem using mathematical induction, but I ...
4
votes
4answers
231 views

Proving that $ \gcd(a,b) = as + bt $, i.e., $ \gcd $ is a linear combination.

For any nonzero integers $ a $ and $ b $, there exist integers $ s $ and $ t $ such that $ \gcd(a,b) = as + bt $. Moreover, $ \gcd(a,b) $ is the smallest positive integer of the form $ as + bt $. I ...
4
votes
3answers
298 views

What is the lowest positive integer multiple of $7$ that is also a power of $2$ (if one exists)?

What is the lowest positive multiple of $7$ that is also a power of $2$ (if one exists)? Not a homework question, I am not in school, I am just wondering what the answer is.
4
votes
6answers
396 views

Prove that $(n-m) \mid (n^r - m^r)$

In respect to a larger proof I need to prove that $(n-m) \mid (n^r - m^r) $ (where $\mid$ means divides, i.e., $a \mid b$ means that $b$ modulus $a$ = $0$). I have played around with this for a while ...
3
votes
3answers
193 views

Demonstração do Teorema de Bezout. (Proof of Bézout's Theorem)

Gostaria de saber como provar usando divisibilidade o teorema de Bezout $(a,b)=d\Longrightarrow \exists f,g\in\mathbb{Z^*}$ tal que $af+gb=d$ I'd like to know how to, using divisibility, ...
3
votes
3answers
400 views

How can I prove that $a|bc$ if and only if $\frac{a}{(a,b)}|c$?

I tried to start by showing that $ \frac{a}{\gcd(a,b)} $ is always an integer, let's call it $d$, because $a$ is always a multiple of $(a,b)$ based on the definition of a g.c.d. I then tried to show ...
3
votes
3answers
786 views

$n^2 + 3n +5$ is not divisible by $121$

Question: Show that $n^2 + 3n + 5$ is not divisible by $121$, where $n$ is an integer.
2
votes
6answers
101 views

Prove that $4^{2n+1}+3^{n+2} : \forall n\in\mathbb{N}$ is a multiple of $13$

How to prove that $\forall n\in\mathbb{N},\exists k\in\mathbb{Z}:4^{2n+1}+3^{n+2}=13\cdot k$ I've tried to do it by induction. For $n=0$ it's trivial. Now for the general case, I decided to throw ...
2
votes
2answers
130 views

7 digit number consisting of 7s and 5s

Find all the 7 digit numbers that have only 5 and 7 as their digits and divisible by both 5 and 7. I have no clue how to use divisibility of 7 to solve this problem. DO i need to check all the 64 ...