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

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The number $2^{3^n}+1$ is divisible by $3^{n+1}$ and not divisible by $3^{n+2}$.

Prove that for every $n\in \mathbb N$, $2^{3^n}+1$ is divisible by $3^{n+1}$ and not divisible by $3^{n+2}$. I was able to prove that $2^{3^n}+1$ is divisible by $3^{n+1}$ using induction. First, ...
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4answers
70 views

How am I supposed to tell if a number is divisible by $13$ (I need a shortcut)?

I've been trying to figure out if a number is divisible by $13$. As I'm saying this in first person, I think I'm supposed to take the rightmost digit of the number, for example, $39$, multiply it by ...
5
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1answer
85 views

Does the A001921 linear recurrent integer sequence always yield composite numbers?

Let $(a_n)$ be the A001921 sequence $$ a_0 := 0,\ a_1 := 7, \quad a_{n+2} = 14a_{n+1} - a_n + 6. $$ Is it true that $a_n$ is always a composite integer for any $n\geq 2$ ? UPDATE : I now make a ...
1
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4answers
101 views

Show that $(k!)^n$ divides $(kn)!$

Show that $(k!)^n$ divides $(kn)!$ I've tried it but without success. Any help would be great.
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2answers
56 views

Prove that, given positive integers m and n, if m | n then 2^m − 1 | 2^n − 1. In particular, deduce that if 2^n − 1 is prime then n is prime.

I think I have the first part of the proof down but I would like to double check that my logic works: m|n $\Leftrightarrow$ n = k*m $\Rightarrow$ $2^n-1 = 2^{km}-1$ ...
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7answers
7k views

If gcd (a,b)=1 and gcd (a,c)=1, then gcd (a,bc)=1

How do I go about proving this? If gcd (a,b)=1 and gcd (a,c)=1, then gcd (a,bc)=1. I'm very confused with gcd proofs.
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1answer
64 views

Under what conditions can $a\sqrt{b} \pm c\sqrt{d}$ be written as $u+v\sqrt{w}$?

Let $a,b,c,d \ge 1$ be integers with $b$ and $d$ nonsquare and $a\sqrt{b} \ge c\sqrt{d}$. Now I have three related questions: Under what conditions can one find $u,v,w$ such that $a\sqrt{b} \pm ...
0
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2answers
92 views

Using inclusion-exclusion principle to count the integers in $\{1, 2, 3, \dots , 100\}$ that are not divisible by $2$, $3$ or $5$

Question: How many integers in $\{1, 2, 3, \dots , 100\}$ are not divisible by $2$, $3$ or $5$? Can anyone give me a full explain of how this applies to the inclusion-exclusin principle? Because I ...
0
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1answer
42 views

Dividing a number into infinite pieces

Last day in physics teacher said that any number divided into infinitely many pieces is zero.It got me thinking in kind of weird direction so here is what I was thinking about and how I tried to ...
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0answers
170 views

Does $A193201$ count the partitions of $n$ of arbitrary dimension?

By my count, these sequences match for $n=0\ldots6$, where partitions that are the same after relabeling dimensions are considered equivalent (i.e., the dimensions are unordered). For example, for ...
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1answer
29 views

Proving that $\varphi(n)$ is divisible by $\varphi(n_1)$ and $\varphi(n_2)$

So, I've been thinking about trying to prove this statement - If $n=n_1n_2$ and $n_1$ and $n_2$ are relatively prime integers greater than 2, prove both $φ(n_1)$ and $φ(n_2)$ divide $φ(n)$. In ...
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0answers
341 views

Is the factorial of the sum of three integers divisible by the product of their factorials? [closed]

Since binomial coefficients are integers, we know that $(x+y)!$ is divisible by $x!y!$ for every nonnegative integers $x,y$. Is it also true that $(x+y+z)!$ is divisible by $x!y!z!$? If so, how to ...
4
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3answers
51 views

divisibility of $n^{15} - n^3$ by $32760$

I have a question & I have no idea where to begin. I hope someone here can help me. Been stuck for a while. Prove or disprove: $n^{15} - n^3$ is divisible by $32760$ for all $n \ge 0$.
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1answer
27 views

Game picking cards so that sum is divisible by $25$

Adele and Bryce play a game. There are $50$ cards, numbered $1,2,\ldots,50$. They take turns alternately picking a card, with Adele going first. If at the end, the sum of the numbers on Adele's cards ...
1
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1answer
63 views

Coprime, commensurable integers

I really need help with proving this problem: For natural numbers k,n > 0 we define set M(k,n) = {k,2k,3k...nk}. Find out which elements are in following sets: a) M(i,n) intersection M(j,n), where ...
5
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1answer
95 views

Find $\gcd$ of two polynomials in $\mathbb{Z}_5[x]$

Question: Find $\gcd$ of $x^4+3x^3 +2x+4$ and $x^2-1$ in $\mathbb{Z}_5[x]$ Applying the Euclidean Algorithm as my book suggests, I got the following: $x^4+3x^3+2x+4=(x^2-1)(x^2+3x+1)+(5x+5)$ ...
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0answers
37 views

Use Euclidean algorithm to find the gcd

$$f(x)=x^3+3x^3+2x+4$$ $$g(x)=x^2+1$$ in $\mathbb Z/5 \mathbb Z[x] $ I got $f(x)=g(x)(x^2+3x+1)+(5x+5)=g(x)(x^2+3x+1)$ as $5x+5->0$ in $\mathbb Z/5 \mathbb Z$, by long division I am not sure how ...
5
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2answers
80 views

Function with $f(a)-f(b)$ dividing $a^3-b^3$

What are all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(a)-f(b)$ divides $a^3-b^3$ for all $a,b\in\mathbb{Z}$ such that $f(a)\neq f(b)$? The constant functions satisfy vacuously, and ...
5
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1answer
74 views

Maximum number dividing $\prod_{i<j}(a_i-a_j)$

Fix an integer $n$. What is the maximum number guaranteed to divide $\prod_{i<j}(a_i-a_j)$ for any integers $a_1,\ldots,a_n$? For instance, if $n=3$, then two of the three numbers have the same ...
2
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3answers
59 views

Where did the sum-of-divisors function come from?

Doing a research project on a few number-theoretic functions, and I was curious, where does the sum-of-divisors function come from? Surely someone thought it up and made it a possibility. I'm talking ...
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2answers
30 views

Find the least $n$ such that the fraction is reducible

So I have this type of question I've never seen before. It smells like Number Theory to me, and I've never studied Number Theory, but I know a very few, very basic Number Theory facts. For instance ...
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0answers
18 views

Axiom of extensionality and Venn diagrams to derive GCD

This is mostly a question of what kind of language to use when explaining the following so as to be rigorous. The wikipedia article on GCD presents a nice intuitive Venn-diagram-based way to derive ...
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1answer
37 views

Prove or disprove: (i) gcd(a,q) = gcd(q,r) (ii) gcd(q,r)|b (iii) gcd(b,r) = gcd(a,q) (iv) gcd(a,r)|q

Given a,b,q,r ∈ ℤ ∋ a = bq + r. Prove or disprove the following: (i) gcd(a,q) = gcd(q,r) (ii) gcd(q,r)|b (iii) gcd(b,r) = gcd(a,q) (iv) gcd(a,r)|q Part (i) is no problem. I'm getting hung up on part ...
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3answers
55 views

Proving a mod b < a/2 when a > b > 0

Suppose that $a \gt b \gt 0$. How can one prove that $a$ mod $b \lt a/2$? I understand why is that happening: if $a$ mod $b \gt a/2$ that means that $a/b \lt a/2$ and $a/b$ has enough "space" to ...
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2answers
233 views

Greatest common divisor power of 6 that divides 73!

Can someone please help me with the following problem? Compute the largest integer power of 6 that divides 73!.
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2answers
32 views

Combining GCD and congruences

Let $a, b, m, k \in \Bbb Z$ such that $m\ge2$ and $k\not=0$. Let $d=\gcd(k,m)$. Prove that if $a\equiv b\pmod m$ and $k$ is a common divisor of $a$ and $b$, then ${\frac ak}\equiv {\frac bk}\pmod ...
2
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1answer
21 views

How do I prove that $R=\{(x,y) \in S \times S : x\text{ divides }y\}$ is antisymmetric?

$S=\{1, 2, 3,\ldots, 1000\}$ $R=\{(x,y) \in S \times S: x \mid y\}$ My attempt: Assume $xRy$ and $yRx$. Then $x=ym$ and $y=xn$ for $m$, $n$ in Natural numbers. -So $x=xxn..$ that gets me nowhere. ...
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1answer
25 views

Reverse a division

I'm working on a program and I'm starting to regret the way I've done this. I start with a user selected number between 0.2 and 24 (lets call it a) then divide 12 by that number (so 12/a = b). Is ...
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2answers
163 views

Congruence class $[a]$ modulo $m$, $\gcd(x, m) = \gcd(a, m)$

I'm currently stumped on this question: Let $a$ and $m$ be integers such that $m\ge1$. Consider the congruence class of $a$, i.e., $[a]$ modulo $m$. Prove that: For all $x\in[a]$, ...
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1answer
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Set of a summation

Let $S = \{n ∈ N | n \text{ divides the sum of any n consecutive numbers} \}$. How can I describe the set S? I was given the hint: $\displaystyle\sum\limits_{n=1}^N n=\frac{N(N+1)}{2}$ I don't want ...
2
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1answer
53 views

Is my understanding right on the divisiblity rule?

For a given number and a divisor. If the prime factors of the divisor can divide a number,then can I say that the divisor will divide a number. For example - 786 divide by 21 If I break 21 in the ...
5
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2answers
109 views

If $a^n-1$ is divisible by $b^n-1$ for all $n$, then $a$ is a power of $b$

Let $a,b$ be natural numbers not equal to $1$ such that $\frac{a^n-1}{b^n-1}$ is natural for any natural $n$. Prove that $a=b^m$ for some natural $m$.
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1answer
15 views

finding A using with restriction $1 \leq a \leq 20$ in GCD

For what $1 \leq a \leq 20$ you are finding $a$ is it true that $a^m+a^n=x^2$ for positive integers $a,m,n,x.$ I did $a^m+a^n=x^2.$ $=a^m(a^{n-m}+1)=x^2$ We know that since $(a,b)=1$ since the ...
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2answers
79 views

$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\dots+\frac{1}{1331}=\frac{p}{q}$; is $p$ divisible by $1997$?

if $p,q\in \mathbb{N}$ and $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\dots+\frac{1}{1331}=\frac{p}{q}$$ why is $p$ divisible by $1997$?
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2answers
79 views

Compute remainder of division

I am trying to compute the remainder of the following division: $$9^{123456789} \quad\textrm{by}\quad 17.$$ Any ideas on how to work this out?
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3answers
75 views

Divisibility by primes

Suppose that $n$ is a natural number, $n \ge 2$, and $n$ satisfies: For each prime divisor $p$ of $n$, $p^2$ does not divide n. If $p$ is prime, $p$ divides $n$ if and only if $(p − 1)$ divides $n$. ...
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3answers
757 views

Does half-life mean something can never completely decay?

Caffeine has a half-life of approximately six hours. I understand this to mean that every six hours, the amount of caffeine in the body is half of what it was six hours prior. Does that mean that ...
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1answer
59 views

Proving property of congruence - help needed

Let $c,d,m,k ∈ \mathbb{Z}$ such that $m ≥ 2$ and $k$ is not zero. Let $f = \gcd(k,m)$. If $c \equiv d \pmod m $ and $k$ divides both $c$ and $d$, then $$ \frac{c}{k} \equiv \frac{d}{k} ...
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0answers
56 views

Proof for $\gcd(F_m,F_n)=F_{\gcd(n,m)}$ [duplicate]

I saw many questions/answers, where: $$\gcd(F_m,F_n)=F_{\gcd(n,m)}$$ is taken as a fact. But how can I actually prove that this is true?
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1answer
80 views

Proving congruence class

Let $a$ and $m$ be integers such that $m ≥ 1$. Consider the congruence class of $a$, $[a]$ modulo $m$. It follows that $∀ x ∈ [a], \gcd(x, m) = \gcd(a, m)$. I have my algebra midterm in two ...
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2answers
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1answer
36 views

Primitive polynomial and divisibility

Let $f(x) \in \mathbb Z[x]$ with $c(f)=1$ and $f$ is non-constant. Now suppose $h(x) \in\mathbb Z[x]$ be such that $h(x)=f(x)q(x)$ where $q(x) \in\mathbb Q[x]$. Then I have to show that $q(x) ...
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2answers
55 views

If for all $n\in\Bbb{N}, a^n-n$ divides $b^n-n$ then $a=b$.

Exercise: Let $a,b\in\Bbb{N}$, show that if for all $n\in\Bbb{N}, \quad a^n-n$ divides $b^n-n$, then $a=b$. I don't have lot of knowledge on this subject, I am aware about some elementary result ...
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2answers
70 views

If $(a^2+b^2) \mid (c^2+d^2)$ and $\gcd(a,b)=\gcd(c,d)=1$ and $\gcd(a,c)>1$, what can be said about the components?

While working on a divisibility problem in integers $a,b,c,d$, with $\gcd(a,b)=\gcd(c,d)=1$, I've come up against the hypothetical condition $$ (a^2+b^2) \mid (c^2+d^2), \tag{$\star$} $$ where, also ...
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2answers
63 views

Find all natural numbers n for which $3^n + 5^n$ is divisible by $3^{n-1} + 5^{n-1}$

Find all natural numbers n for which $3^n + 5^n$ is divisible by $3^{n-1} + 5^{n-1}$. Interested if there is a nice quick way other than mine.
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1answer
59 views

Greatest Common Divisor of two binary polynomials

How can I find the GCD of $x^4 + x^3 + x^2 + 1$ and $x^6 + x^5 + x^4 + x^3 + x^2 + 1$? I know that $x^4 + x^3 + x^2 + 1$ is an irreducible polynomial of degree $4$, and that it is not primitive, but ...
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5answers
123 views

Show that $\gcd(a,b)>1$

Given are three natural numbers $a$, $b$ and $c$, for which $$\frac1a+\frac1b=\frac1c,$$ show that $\gcd(a,b)>1$. Could you someone provide a hint? I already tried algebraic manipulation, but ...
0
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1answer
46 views

If a|b and b|a, find the value of a in terms of b.

If a|b and b|a, where a and b are integers and a≠0, find the value of a in terms of b. Assume that b>0.
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2answers
112 views

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$.

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$. Here $n\in\mathbb N$, $a,b,c,x,y,z\in\mathbb Z$. This problem is originally ...
0
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2answers
34 views

For any integer a, if $6|(3−a)$, then $3| (a−2)$. [closed]

Prove: For any integer a, if $6|(3−a)$, then $3| (a−2)$. I've been trying to work this problem for a while, but missed a day of class and can't seem to work it out.