Tagged Questions

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

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1
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1answer
30 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 ...
1
vote
2answers
20 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 ...
0
votes
0answers
10 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 ...
1
vote
0answers
30 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 ...
0
votes
1answer
26 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 ...
1
vote
3answers
32 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 ...
3
votes
0answers
51 views

What is the most elementary way of proving a sequence is free of non-trivial squares?

Given the sequence A001921 $$ 0, 7, 104, 1455, 20272, 282359, 3932760, 54776287, 762935264, 10626317415, 148005508552, 2061450802319, 28712305723920, 399910829332567, \dots $$ which obeys the ...
-3
votes
2answers
106 views

Compute largest integer power of $6$ that divides $73!$ [on hold]

I am looking to compute the largest integer power of $6$ that divides $73!$ I need to show working out also. Any help or hints appreciated
0
votes
2answers
22 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
votes
1answer
13 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. ...
1
vote
1answer
21 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 ...
0
votes
2answers
190 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!.
1
vote
3answers
36 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 ...
0
votes
2answers
96 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 $x\in[a]$, ...
2
votes
1answer
46 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 ...
0
votes
1answer
13 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 ...
7
votes
0answers
40 views

Prove or disprove $\gcd(q,r) \mid b$ if $a = bq + r$

Prove or disprove $\gcd(q,r) \mid b$ if $a, b, q, r \in \Bbb{Z}^+ \ni a = bq +r$ I'm pretty sure it's true (I can't think of a counter example), but I don't see how to prove it. Some of my ...
3
votes
2answers
75 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 $1997$?
0
votes
2answers
72 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?
1
vote
2answers
57 views

Use Fibonacci number to prove that is the integer that is closest to another number

Hi everyone, I don't really understand the problem. I have the following hint, but I don't know how to work it.
0
votes
0answers
27 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?
0
votes
1answer
41 views

Prove, that if $a$ and $b$ are natural numbers and $(4a^2 -1)^2$ is divisible by $4ab - 1$ then $a = b$ [on hold]

Prove that if $a$ and $b$ are natural numbers and $(4a^2 -1)^2$ is divisible by $4ab - 1$ then $a = b$
0
votes
1answer
53 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 ...
0
votes
1answer
40 views

Modulus proof with gcd [closed]

Let $a,b,m,k\in\Bbb Z$ such that $m\ge2$ and $k\ne0$. Let $d=\gcd(k,m)$. Prove the following: If $a\equiv b\bmod m$ and $k$ is a common divisor of $a$ and $b$, then $\left(\dfrac ...
0
votes
1answer
42 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} ...
4
votes
2answers
49 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 ...
0
votes
2answers
21 views

Proof involving greatest common divisor [closed]

Suppose that $\text{gcd}\:(a, y) = 1$ and $\text{gcd}\:(b, y) = d$. How do I show that $\text{gcd}\:(ab, y) = d$?
1
vote
2answers
45 views

How can I prove that the square root of two prime numbers multiplied is non-rational number?

$P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?
0
votes
2answers
49 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.
-1
votes
1answer
28 views

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 ...
0
votes
1answer
37 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 ...
3
votes
2answers
56 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 ...
0
votes
2answers
32 views

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

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.
2
votes
3answers
22 views

Prove if a|c and b|d and gcd(c,d)=1 then gcd(a,b)=1

I'm trying to prove that if a|c and b|d and gcd(c,d)=1 then gcd(a,b)=1 So far, I have assumed that: Since gcd(c,d) = 1 then by EEA, gcd(c,d) = 1 = cx + dy for some x,y that are integers. And since ...
2
votes
2answers
71 views

Probability a product of $n$ randomly chosen numbers from 1-9 is divisible by 10.

I'm working on a problem where each number is chosen randomly from 1-9. Given $n$ numbers chosen in this manner, we multiply all of these together. I'm looking for the probability that this product is ...
0
votes
0answers
28 views

Equation with gcf , lcm

Can you please help me with this? I have no idea how to solve this problem Find all positive integers $a$, $b$ such that $$a+b+\gcd(a,b)+\text{lcm}(a,b)=50$$ Thank you for answer
1
vote
1answer
21 views

if $p=(a+ib)(c+id)$ and $p^2 = a^2 + b^2$ then $p\mid a$ & $p\mid b$

We're working on Gauss integers... p is an odd prime such that $p \not\equiv 1 \pmod 4$. We want to prove that if there is $(a,b,c,d) \in \mathbb{Z}^4$ such that $$p = (a+ib)(c+id) \text{ ...
1
vote
0answers
16 views

Using divisibility and greatest common divisor for a proof

If u|t and v|t and gcd(u,v)=1, then prove that (uv)|t I started by analyzing the definition of divisibility and I got that (uv)|t^2, but this doesn't help me. Any advice would be appreciated. Thank ...
0
votes
1answer
41 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.
0
votes
1answer
45 views

Is it possible to find $n-1$ consecutive composite integers

Given an integer $n\geq 2$ ,can we always find an integer $m$ such that each of the $n-1$ consecutive integers $m+2,m+3,.....,m+n$ are composite?
1
vote
2answers
42 views

Which of the following numbers does not divide $2^{1650}-1$?

I'm practicing for a math competition that is coming up, and I got stuck on this question: Which of the following numbers does not divide $2^{1650}-1$? $3$, $7$, $31$, $127$, $2047$ I've seen a ...
1
vote
2answers
26 views

Algorithm to find the coefficient of GCD linear combination?

One of the properties of the GCD of two integers is that it can be written as the linear combination of the two, is there an algorithm that can be used to find the coefficients of this linear ...
3
votes
5answers
80 views

How to prove that $8^{18} - 1$ is divisible by $7$ [duplicate]

How to prove that: $$ 8^{18}-1\equiv0\pmod7 $$ In the simplest way?
0
votes
0answers
16 views

Proof with GCD(m, n) [duplicate]

How can I prove that equation below is true? If $a > b$ and $a, b$ are relatively prime numbers, then for $0 <= m < n$: $GCD(a^n - b^n, a^m - b^m) = a^{GCD(m,n)}-b^{GCD(m,n)}$
1
vote
0answers
20 views

GCD-Domain and proprieties

Let $A$ be a commutative GCD-domain (not necessary UFD or Bezout) and $a,b,c$ elements of $A$ such that $\gcd(a,b) = \gcd(b,c) = \gcd(a,c) = 1$. Is it true that $\gcd(ab,c) = 1$ ?
-1
votes
0answers
25 views

Range divisibility of large numbers

Question: Consider a range of positive integers from $[L,R]$ and a set of other positive integers $A = \{\ldots\}$. Find the number of integers in $[L,R]$ that are divisible by any of the ...
2
votes
1answer
29 views

The greatest common divisor is the smallest positive linear combination

How to prove the following theorems about gcd? Theorem 1: Let $a$ and $b$ be nonzero integers. Then the smallest positive linear combination of $a$ and $b$ is a common divisor of $a$ and $b$. ...
1
vote
1answer
32 views

Something similar to Euler's theorem

If $p$, $q$ are not equal primes. $n=pq$, $\varphi(n) = (p − 1)(q − 1)$, $d = \gcd(p − 1, q − 1)$. Is it true that for any $a$ such that $\gcd(a, n) = 1$ holds $a^{\frac{\varphi(n)}{d}} \equiv 1 ...
1
vote
1answer
24 views

How can I show that the following number is not divisible by $p$ prime?

Let $p$ be a prime number. Let $k$ be some natural number and $r$ be some nonnegative integer. Then, I want to show that for $1\leq i\leq p^k-1$, \begin{equation*} \frac{p^{k+r}m-i}{p^k-i} ...
2
votes
3answers
48 views

How to prove that $\gcd(2n+3, 3n+1)$ divides $7$?

How can I start proving that gcd(2n+3, 3n+1) | 7? EDIT: It is $\gcd(2n+3, 3n+1)$ divides $7$. My bad. Thanks paw88789.