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

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2answers
28 views

Number 9 and age of mother when child is born.

If a mother's age is divisible by 9 when a child is born then once you go to the next decade,n every 11 years the child's age and mother's age are always the same two numbers in reverse order. For ...
6
votes
5answers
172 views

If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$

Let $ a,b,c$ positive integer such that $ a + b + c \mid a^2 + b^2 + c^2$. Show that $ a + b + c \mid a^n + b^n + c^n$ for infinitely many positive integer $ n$. (problem composed by Laurentiu ...
1
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0answers
16 views

Lehmer's totient problem generalization (adding a constant )

Lehmer's conjecture is an well-known open problem which states that the divisibility : $$\phi(n) \mid n-1$$ holds only for primes . This motivated me to ask the following : For which ...
1
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2answers
31 views

Prove $(a, b) \mid ((a + b), (a - b))$

I tried this: Suppose $(a, b) = d$. Then $ax + by = d$. Let $((a + b), (a – b)) = e$. Then $$\begin{align}e& = (a + b)u + (a – b)v\\ &= au + bu + av – bv\\ &= a(u + v) + b(u – ...
2
votes
3answers
92 views

If $\gcd(ab,c)=d$ and $c|ab$ then $c=d$

For all positive integers $a$, $b$, $c$ and $d$, if $\gcd(ab, c) = d$ and $c | ab$, then $c = d$. Need help proving this question, I know that $abx + cy = d$ for integers $x,y$ and that $c|ab$ can be ...
0
votes
0answers
12 views

Divisibility proof with GCD condition

Suppose $a|m$, $b|m$ and $\gcd(a,b) = 1$. Prove, without appealing to the fundamental theorem of arithmetic, that $ab|m$. I know that $\gcd(a,b)=1$ means they are relatively prime. I also know ...
1
vote
2answers
120 views

Divisibility and GCD proof

I'm having trouble with this simple proof. Any help would be appreciated. I don't really know where to start to try to conquer this problem. Suppose $a|m$, $b|m$ and $\gcd(a,b) = 1$. Prove, ...
0
votes
1answer
18 views

If $a, b \mid c \text { and } \gcd(a, b) = d, \text { then } ab \mid cd $

$a \mid c \to c = ak \text { and } b \mid c \to c = bj.$ $ak + bj = 2c = d \to c \mid d.$ $d \mid a \to a = dj.$ $c = ak = d(jk) \to d \mid c.$ So, $c = d.$ $a \mid c \text { and } b \mid c ...
2
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0answers
25 views

About the least common multiple of numbers and combinatorial

Prove that for any positive integer $n$, the least common multiple of the numbers $1, 2, 3, \ldots , n$ and the least common multiple of the numbers: ${n\choose 1}, {n\choose 2}, \ldots , {n\choose ...
2
votes
1answer
104 views

Difficult sets of Equations, counting

Let $ m$ be the number of solutions in positive integers to the equation $ 4x+3y+2z=2009$, and let $ n$ be the number of solutions in positive integers to the equation $ 4x+3y+2z=2000$. Find the ...
-3
votes
3answers
56 views

Prove that: 1. $gcd(a,b)=lcm(a,b)$ iff $|a|=|b|$ 2. $k>0\implies lcm(ka,kb)=k lcm(a,bk)$ 3. $a\mid m, b\mid m$, then $lcm(a,b)\mid m$

Let $a,b$ any non-zero integers. Prove that: $gcd(a,b)=lcm(a,b)$ If and only if $|a|=|b|$. If $k>0$, then $lcm(ka,kb)=k lcm(a,bk)$ if $m$ is multiple of $a$ and $b$, then $lcm(a,b)$ divides $m$ ...
7
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0answers
124 views

Dividing the whole into a minimal amount of parts to equally distribute it between different groups.

Suppose we have a finite amount of numbers $x_1, x_2, ..., x_n$ ($x_i\in\mathbb{N}$) and an object that should be divided into parts in such a way that it can be without further dividing distributed ...
5
votes
1answer
55 views

Missing values of the ratio $\frac{(x+y+z)^2}{x^2+y^2+z^2}$

Let $x,y,z$ be some positive integers. Is it true that we cannot find any positive integer $n$ for which $$ \frac{(x+y+z)^2}{x^2+y^2+z^2}=1+\frac{2}{3n}\,\,? $$
2
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2answers
63 views

For what values of $n$ , does $7 \mid 5^n+1$

$7 \mid 5^n+1$ implies $5^n+1=7a$ for some integer $a$ i.e $5^n=7a-1$ Now , $5^n$ is an integer which always ends with $5$ [for any integer $n$]. Thus , $7a-1$ must also end with $5$.But , this is ...
5
votes
0answers
46 views

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have?

A natural number has exactly 10 divisors including 1 and itself.How many distinct prime factors can this natural number have? options given: (a) either $1$ or $2$ (b)$1$ or $3$ (c)either $2$ or ...
0
votes
4answers
47 views

Why Zero divided by Zero is undefined and not Infinity [duplicate]

apologize in advance if this is a duplicate, but I found a lot questions related to this but none answering this specific question. My logic is: let's consider division the opposite of ...
3
votes
3answers
140 views

when ${\rm gcd} (a,b)=1$, what is ${\rm gcd} (a+b , a^2+b^2)$? [duplicate]

I want to prove above statement "what is ${\rm gcd} (a+b , a^2+b^2)$ when ${\rm gcd}(a,b) = 1$" I've seen some proofs of it, but i couldn't find useful one. here is one of the proof of it. some ...
1
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2answers
43 views

$z=100^2-x^2$. Then, how many values of $x,z$ are divisible by $6$?

$z=100^2-x^2$. Then, how many values of $x,z$ are divisible by $6$? My approach: For $x=1$, $z$ is not divisible by $6$. For $x=2$, $z$ is divisible by $6$. For $x=3$, $z$ is not divisible by ...
0
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2answers
52 views

How to find $\frac{a+b+c}x$? [closed]

$ab$ and $bc$ are two digit numbers. if $ab*x=2 $ and $bc*x=3$ then find $\frac{a+b+c}x$. (* is multiplication) It looks simple but I couldnt go further. $$17b=2(15a-c)\iff b\mid2 \quad and\quad ...
6
votes
8answers
353 views

If a prime $p\mid ab$, then $p\mid a$ or $p\mid b$

If a prime number $p$ is a divisor of a product $ab$, $p$ has to be a divisor of $b$ or $a$. How can I demonstrate this theorem? I demonstrated this theorem on one way using Bezout's theorem in an ...
6
votes
6answers
4k views

Prove $\gcd(a+b, a-b) = 1$ or $\gcd(a+b, a-b) = 2$ if $\gcd(a,b) = 1$

I want to show that for $\gcd(a,b) = 1$ $a,b \in Z$ $\gcd(a+b, a-b) > = 1$ or $\gcd(a+b, a-b) = 2$ holds. I think the first step should look something like this: $d = \gcd(a+b, a-b) ...
1
vote
3answers
905 views

Show that the gcd of an odd integer and an even integer is odd

I am using the definition of odd and even integers along with bezout's theorem and I end up with something of the form $d=(2k)m+(2l+1)p$ where $a=2k$ and $b=2l+1$. I've tried to use contradiction as ...
3
votes
0answers
63 views

Connections between Fibonacci and natural numbers

Here are some known facts about Fibonacci numbers and then some questions regarding them . 1.Carmichael's theorem . For every $n>12$ $F_n$ has a prime divisor which doesn't divide any of the ...
3
votes
1answer
66 views

Each number in a subset $S\subseteq \{1,\ldots,2n\}$ does not divide another one. Then $\max |S|$?

This problem comes from a seemingly innocuous question from a professor during a lesson for a Math Olympiad course. [A part of this question is really a classic of number theory/combinatorics] ...
3
votes
3answers
74 views

Prove that $\sqrt{3}$ is not a rational number [duplicate]

There is a similar question however that question asks why $3 |p^2$. Here the question is about $ 3 | p^2 \rightarrow 3 | p$. It is a simple exercise (1.2.1) from Abbot's "Understanding Analysis". ...
2
votes
2answers
47 views

Proof that: $a=bq+r ,-\frac {|b|}{2}<r≤\frac {|b|}{2}$ [closed]

Proof that: Let $a,b$ any integers, with $b≠0$, Then there exist unique integers $q$ and $r$ surch that $$a=bq+r ,$$ where $$-\frac {|b|}{2}<r≤\frac {|b|}{2}$$ Note corolario: Let ...
1
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1answer
13 views

Proving with divisibility

I have never written any proofs (except high school geometry) in my life, so I'm not sure what exactly the proper formatting should be. Involving divisibility, the proposition states: Let $a, b,$ ...
3
votes
5answers
374 views

What does “$x$ divides $y$” mean?

I need to negate the following sentence: "If for the integers $x, y, z$ we know that $x$ divides $y$ and $y$ divides $z$, then $x$ divides $z$." In this scenario, what does it mean for $x$ to ...
1
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1answer
17 views

$t > 0 $ is the least common multiple of $a, b$ (not both $0$) iff $a, b \mid t$ and $a, b \mid c \to t \mid c$

My attempt: Suppose $[a, b] = t =$ lcm of $a, b.$ By definition of lcm $a, b \mid t$. If $a, b \mid t$ and $a, b \mid c$, then $|t| \le |c|$ since $t$ is the smallest such integer. So, $t \mid c$. ...
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votes
2answers
81 views

Direct proof divisibility: Suppose $x$ is an integer such that $2 \cdot 3 \cdot 4 \cdot 5 \cdot x = 59 \cdot 58 \cdot 57 \cdot 56 \cdot 55$

Suppose $\,x\,$ is an integer such that $\,2 \cdot 3 \cdot 4 \cdot 5 \cdot x = 59 \cdot 58 \cdot 57 \cdot 56 \cdot 55.\,$ Does $\,59 \mid x$? Does $\,29 \mid x$? Does $\,118 \mid x$?
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0answers
20 views

formula to find the lowest whole number divisible by two other numbers? [closed]

does anyone have a formula to find the lowest whole number divisible by two other numbers? Thanks Jo
1
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2answers
41 views

Guessing how many times a smaller number goes into bigger number

For example when diving 105 / 148. After you add a number 0 to the numerator, the division becomes 1050 / 148. The answer becomes a decimal with 1050 / 148. The two numbers are not divisible by a ...
9
votes
2answers
142 views

Can the identity $ab=\gcd(a,b)\text{lcm}(a,b)$ be recovered from this category?

Define the category $\mathcal{C}$ as follows. The objects are defined as $\text{Obj}(\mathcal{C})=\mathbb{Z}^+$, and a lone morphism $a\to b$ exists if and only if $a\mid b$. Otherwise ...
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votes
4answers
160 views

How can I find The Multiplicative Inverse of $1+\sqrt{2}$? [closed]

I am doing contemporary abstract algebra and am working in an integral domain. I have found it necessary to compute the multiplicative inverse of $1+\sqrt{2}$; I know such the definition of a ...
3
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1answer
74 views

Irrationality of ${5^{1/7}}$

I am struggling with elementary proofs, and would appreciate any feedback as to the logic and structure of my work. Show that ${5^{1/7}}$ does not represent a rational number. Suppose ${5^{1/7}}$ is ...
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1answer
22 views

Modular Arithmetic Divisibility

Prove that for all integers $n$, exactly one of $n$, $2n − 1$ and $2n + 1$ is divisible by $3$.
2
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1answer
45 views

Prove that it is NOT true that for every integer $n$, 60 divides $n$ if and only if 6 divides $n$ and 10 divides $n$.

This is Velleman's exercise 3.4.26 (b): Prove that it is NOT true that for every integer $n$, 60 divides $n$ iff 6 divides $n$ and 10 divides $n$. I do understand that a number will be ...
3
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2answers
58 views

If $2xy$ divides $x^2+y^2-x$, prove that $x$ is a perfect square [duplicate]

This problem is from ( BMO Exam1991 ). I tried to solve but it was difficult. The problem is: If $ x^{2} + y^{2} - x $ is a multiple of $ 2xy $ where $x$ & $y$ are integers, prove that $x $ ...
4
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2answers
44 views

Is there a Divisibility Metric for Numbers?

Both prime numbers and highly divisible numbers have a common characteristic: divisibility. The former are divisible by as few lower numbers as possible, and the latter by as many as possible, like ...
4
votes
3answers
83 views

Looking for an example of a GCD domain which is not a UFD

I know that every UFD (unique factorization domain) is a GCD domain i.e. g.c.d. of any two elements, not both zero, exists in the domain. I am looking for an example of a GCD domain which is not ...
3
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0answers
30 views

Is there a name for the least exponent $e$ such that a power of an integer is divisible by another?

Say the primes dividing $m$ also divide $n$. Is there a name for the least exponent $e$ such that $m | n^e$? I can write that explicitly using the prime factorizations of $m$ and $n$, but am ...
1
vote
1answer
11 views

$\gcd(ca,cb)\mid ca$ and $\gcd(ca,cb)\mid cb \to \gcd(ca,cb)\mid cd$.

Let $(ca)x + (cb)y = cd$ where $d = (a, b).$ Then since $\gcd(ca,cb)\mid ca$ and $\gcd(ca,cb)\mid cb \to \gcd(ca,cb)\mid cd$. I don't get how they deduced the conclusion. For one thing, ...
3
votes
2answers
34 views

$\gcd (ca, cb) = \gcd (a, b)c$ if $c > 0$

Let $\gcd (a, b) = d$. So, $ax + by = d$ for some $x, y$. Then $(ca)x + (cb)y = cd$. Thus, $\gcd (ca, cb) = cd = \gcd(a, b)c$. Does it work?
4
votes
3answers
287 views

Find all integers such that $2 < x < 2014$ and $2015|(x^2-x)$

Find all integers, $x$, such that $2 < x < 2014$ and $2015|(x^2-x)$. I factored it and now I know that $x > 45$ and I have found one solution so far: $(156)(155)= (2015)(12)$. It's just that ...
3
votes
1answer
139 views

Prove that $n \in \mathbb{Z}^\star \Leftrightarrow n \mid 1$ and $n-1 \mid 1$ or $n+1 \mid 1$, and $(x-1)/(t-1) \equiv n \pmod {t-1}$

I want to prove the following lemma: Let $F[t,t^{-1}]$ be the ring of the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$ and assume that the characteristic of $F$ is zero. ...
3
votes
2answers
41 views

If $p$ and $q$ are primes, which binomial coefficients $\binom{pq}{n}$ are divisible by $pq$?

If $p$ and $q$ are primes, which binomial coefficients $\binom{pq}{n}$, $1 \le n < pq$, are divisible by $pq$? In particular, if $p$ and $q$ are distinct odd primes, and $n$ is even, does $pq ...
1
vote
1answer
34 views

Permutations where no partial sum is divisible by 3 (contest question)

A permutation of the integers $1901,1902\dots 2000$ is a sequence in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums $$s_1 = ...
1
vote
1answer
34 views

Divisibility proofs for greatest common divisor

I am studying divisibility and greatest common divisors. I have reached a section where I need to prove properties. My question is: are my proofs substantial? Or do I need to add to them? Below are ...
0
votes
1answer
29 views

Is this assertion about g.c.d. true? [closed]

Is it true that if $\gcd(a,bc)=1$ and $\gcd(b,c)=1$ then $\gcd(a,b^2)=\gcd(a,c^2)=\gcd(ab^2,c^2)=\gcd(a,(bc)^2)=1$? Many thanks.