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

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Proof $\forall n \in \Bbb N$ that $2^n \cdot \prod_{i = 1}^{n} (2i-1)$ is divisible by $n!$

I'm trying to prove it by induction. $P(1)$ holds true. My inductive hypothesis is $n!\ |\ 2^n \frac {2n!} {2^n n!}$ which simplifies to $n!\ |\ \frac {2n!} {n!}$. Next $P(n+1)$: $$(n+1)!\ |\ 2^{n+1} ...
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0answers
10 views

Integer division and congruence exercise

I'm just starting with integer division and congruence in an algebra course and I have this problem: Let $a$ be an odd integer. Prove that $\forall n \in \Bbb N$: $$2^{n+2}\ |\ a^{2^n} - 1$$ I've ...
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10answers
9k views

Has there ever been an application of dividing by zero?

Regarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), and ...
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0answers
17 views

Linear factor divides a function

I just came up with a simple question. If I have a polynomial function $f(x_1,x_2,\ldots,x_n)$ and I know that when $x_i=x_j, f=0$. Then does it imply $x_i-x_j$ divides $f$ for all $i\neq j$? If yes, ...
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1answer
15 views

Help with congruence and divisibility exercise

I'm starting to solve some problems of congruence and integer division, so the exercise is quite simple but I'm not sure I'm on the right track. I need to prove that the following is true for all $n ...
2
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2answers
56 views

$7^{6} | (a+b+ab)^2$ Find the value of $a,b$ [on hold]

$7^{6} | (a+b+ab)^2$ Find the value of a,b. I have used trial and error for a singular solution. But a generalized solution will be helpful. Provide me the concept to deal with this problem and ...
2
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1answer
34 views

Find all $n$ such that $n|1^n + 2^n + 3^n + \cdots + (n-1)^n$ where $n \in \mathbb{Z}^+$.

Find all $n$ such that $$n|1^n + 2^n + 3^n + \cdots + (n-1)^n$$ where $n \in \mathbb{Z}^+$. I don't know how to start. $n = 3, 5$ are simple solutions. Induction seems strange since the divisor ...
2
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2answers
77 views

cancelling out before evaluation of variable

I'm been working on a theory, though my math is weak. Let's say I've managed to determine that I can arrive at an answer A by always using the formula BCD / D. Of ...
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1answer
2k views

Formula of MIPS (million instructions per second)

Could you please help me to understand the mathematics behind MIPS rating formula? The performance of a CPU (processor) can be measured in MIPS. The formula for MIPS is: $$\text{MIPS} = ...
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1answer
15 views

Find elements of a set that divide an expression.

I have to determine the elements of the following set: $A = \{x\in\ \mathbb Z \vert \sqrt[3]{\frac {7x + 2}{x+5}} \in \mathbb Z \}$ I know that $x+5 \not=0$ and $x+5$ must divide $7x + 2$ but I ...
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1answer
35 views

Divisibility by 11 [on hold]

$xyzw$ is a 4 digit number such that x is even,y is smallest possible odd number,z is greatest prime no,w is a multiple of 3.if ($xyzw + n$) is divisible by 11 then n cannot be: a. 6711 ...
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0answers
26 views

Divisibility of N [on hold]

If there are $N$ tuples $(a,b,c)$ such that they are positive integers and $a>b>c$ and $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2}$$ then $N$ is divisible by ? The answer is an integer ...
4
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3answers
103 views

$\frac{2abc}{(a+b-c)(b+c-a)(c+a-b)}$ a positive integer

Find all triplets $(a,b,c)$ of positive integers so that $\gcd(a,b,c)=1$ and $$ \frac{2abc}{(a+b-c)(b+c-a)(c+a-b)} $$ is a positive integer. What I've done: first I looked with Mathematica for ...
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0answers
14 views

Divide a number by two different numbers in combination

Similar to this question: How to divide a number by $2$ numbers? I would like to know how to divide a number (let's say 23) by two different numbers (say, 1.6 and 2.4) to find what combination of the ...
3
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2answers
254 views

How to understand Apostol's proof of the irrationality of $\sqrt{n}$ if $n$ is not a perfect square?

Recently I am reading the textbook of Apostol, Mathematical Analysis, Second Edition. On page 7, there is a theorem 1.10: If $n$ is a positive integer with is not a perfect square, then $\sqrt{n}$ is ...
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1answer
18 views

Find a unique value for $d$ in $(d \cdot e) \pmod{F} \equiv 1$

Given that I know the value of $e$ and $F$. How to determine an unique integer value for $d$ in such a way that the reminder of the division of $(d \cdot e)$ per $F$ is equal to one? $(d \cdot e) ...
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0answers
41 views

Prime number theory. [closed]

If $a$ is coprime to $b$ and $y$ and $b$ are both coprime to $x$; then Prove that $ax+by$ is a coprime to $ab$.
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1answer
38 views

Solving $(ap)^2-d(bq)^2=1$ for distinct primes $p,q$

I'm pondering the following claim regarding special cases of the Pell equation. Conjecture: For every pair of distinct primes $p$ and $q$, there exist integers $a$ and $b$, and a non-square integer ...
3
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0answers
277 views

The Divisors of $s(2s+1)$ and Primes $2n+1$ and $3n+1$ part 1

I want to check my math (and proof) on the following claim. The claim is by way of a computer search and a "hunch". claim: If $s$ is a prime number I write $\varphi_{s} =s(2s+1)$. Let $\tau$ be ...
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1answer
37 views

Number theory, prove that a prime number $p \mid 1$

Consider a prime number $p > 1$ and $a \in \mathbb{Z}$ and $p < a$. We know $p \mid a$, then $a = p.b$ for $b \in \mathbb{N}$. We also already know the congruence $a \equiv 1 (\text{mod } m)$ ...
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2answers
23 views

Let $n = 2^{31}*3^{19}$. Find the number of positive divisors $d$ of $n^2$ such that $1\leq d\leq n$ and $d \nmid n$

Let $n=2^{31}*3^{19}$. Find the number of positive divisors $d$ of $n^2$ such that $1\leq d\leq n$ and $d$ does not divide $n$. My attempt $n^2 = 2^{62} * 3^{38}$ Total divisors $= 1 + 62 + 38 + ...
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1answer
74 views

A seemingly-trivial divisibility conjecture

While working on another problem, I stumbled on the following divisibility claim. Conjecture: No integers $a,b,c,d$ satisfy all of the following conditions: $a^2+b^2-c^2-d^2 = 2(ad-bc)-1$; ...
0
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0answers
29 views

If $k(a^2+mb^2) = c^2+md^2$, what can be said about the form of $k$?

Let $k,a,b,c,$ and $d$ be integers, and let $m \ge 2$ be a non-square integer, such that $$ k(a^2+mb^2) = c^2+md^2. $$ QUESTIONS: What can be said about the form of $k$ with no further ...
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2answers
27 views

Divisibility of Exponents

So I'm having trouble trying to show this, a,b and x are positive integers. If $a\mid b^x$, show that some factor $k$ of $a$ divides $b$. In other words, if a number $a$ divides a power, how can I ...
3
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3answers
54 views

Find out all solutions of the congruence $x^2 \equiv 9 \mod 256$.

I need to find all the solutions of the congruence $x^2 \equiv 9 \mod 256$. I tried (apparently naively) to do this: $x^2 \equiv 9 \mod 256$ $\Leftrightarrow$ $x^2 -9 \equiv 0 \mod 256$ ...
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5answers
79 views

Proof that if $(n+1)^2 -1$ is even then $n$ is even?

The forward implication, if $n$ is even then $(n+1)^2 -1$ is even, was simple. I can't figure out the other implication: if $(n+1)^2 -1$ is even then $n$ is even. What type of proof do I want to ...
3
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3answers
272 views

Which Digit-Permutations Preserve Divisibility?

This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it. Let $n > 1,b > 1$ be integers and ...
2
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1answer
32 views

Fraction simplification Rules

I am studying for GRE and One of the practice questions is a division. After converting my Mixed numeral I get 90/72 now I just have to simplify. What I understood is that you divide by Least common ...
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3answers
280 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 ...
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2answers
35 views

GCD divisibility of LCM

Show that the following conditions are equivalent: i) There exist positive integers $a,b$ such that $\gcd(a,b)=d$ and $\operatorname{lcm}(a,b)=m$. ii) $d∣m$ The first direction is very ...
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2answers
41 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 – ...
3
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1answer
397 views

$2^a +1$ is not divisible by $2^b-1$.

Let $a,b>2$ be positive integers. We need to show that $2^a +1$ is not divisible by $2^b-1$. Could any one give me hint?
2
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3answers
53 views

For $a,b>2$, $a,b\in \Bbb{N}$ , prove that $2^a+1$ is never divisible by $2^b-1$ [duplicate]

I have to prove that for $a,b>2$, $a,b\in \Bbb{N}$ that $2^a+1$ is never divisible by $2^b-1$. The method I used is by taking cases, first of them being $b>a$. Now since $b>a$ implies ...
0
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1answer
46 views

If $\ (1+3x+x^2)^{10}=\sum_{r=0}^{20}a_r x^r\ $ then…

If $$\ (1+3x+x^2)^{10}=\sum_{r=0}^{20}a_r x^r\ $$ Then then what is the least number except 1 which divides the following:$$\ \sum_{r=0}^{20}(3r+1)a_r\ $$ EDIT: i have put x=1 then it is something ...
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3answers
68 views

Without using modular equivalence, show that: $\gcd(4n^2+1,24)=1$

Without using modular equivalence, show that: $\gcd(4n^2+1,24)=1$ Let $d=\gcd(4n^2+1,24)$ then we have: $$d|24n^2+6,24n^2\ \Rightarrow\ d|6\ \Rightarrow\ d|6n^2,4n^2+1\ \Rightarrow\ d|12n^2,12n^2+3\ ...
5
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2answers
38 views

For any $a$ in $\Bbb Z$, prove that $6|a(a+5)(a+10)$

So I am given this question for my number theory and proof class: For any $a \in \Bbb Z$, prove that $6|a(a+5)(a+10)$. I've thought about a few different ways to approach this. I think I could ...
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14answers
34k views

Dividing 100% by 3 without any left

In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left. Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quality'. The totality ...
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3answers
45 views

How many numbers less than 100 have the sum of factors as odd?

How many numbers less than 100 have the sum of factors as odd? Answer is 16 This question and explanation is taken from careerbless.com The link given derives the answer using some ...
0
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3answers
62 views

Show that among every consecutive 5 integers one is coprime to the others

Show that among every consecutive 5 integers one is coprime to the others I considered these 5 numbers as: $5k,5k+1,5k+2,5k+3,5k+4$ It's seen that for example $5k+1$ is coprime to $5k$ and $5k+2$,now ...
2
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1answer
29 views

Prove that n is a multiple of four…

Let $a_1, a_2, a_3,....a_n$be $n$ numbers such that $a_i$ is either $+1$ or $-1$. If $a_1a_2a_3a_4 + a_2a_3a_4a_5 +...+a_na_1a_2a_3=0$, then prove that $4$ divides $n$. Well $2$ definitely divides ...
2
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3answers
276 views

Proof using deductive reasoning

I need to deductively prove that the sum of cubes of $3$ consecutive natural numbers is divisible by $9$. I can prove deductively that they are divisible by $3$ but so far any combination I choose ...
2
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1answer
52 views

Finding the integers

Find all integers $a,b,c$ with $1<a<b<c$ such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$. I cannot understand how to solve this. I would appreciate any help.
2
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2answers
94 views

Is it divisible by $3^n$?

I need to prove that a number made up exactly $3^n$ $1$s and nothing else is a multiple of $3^n$. Well I think it is true that any number is a multiple of $3^n$ if the sum of its digits is. But I ...
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0answers
18 views

Get first digits of a very large quotient

Is there a method to get the first $n$ digits of a quotient (ex. a thousand digit number divided by a 5 digit number) without dividing all the way through? I suppose long division until $n$ digits are ...
3
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0answers
64 views

When does $a^b\mid b^a$

Let $a,b >1$ be integers. When does $a^b \mid b^a$? Certainly if this is true then $a\mid b$ by considering $a$'s prime factors. (not quite convinced). Also then if $b$ is prime then $a=b$. ...
6
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2answers
59 views

How to prove $p^2 \mid \binom {2p} {p }-2$ for prime $p$?

How to prove $p^2 \mid \binom {2p} {p } -2$ for prime $p$? I have a hint: for $1 \le i \le p-1$, $p \mid \binom p i$. I cannot even start the proof. Please help.
0
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1answer
18 views

Need an assistance with a specific step of a specific Division Algorithm proof

I'm trying to wrap my head around a Division Algorithm's proof. That is, Let $a, b \in \mathbb{Z}, a \neq 0$. Then there are unique $q,r \in \mathbb{Z}$ such that $b = qa + r, 0 \leq r < |a|$. ...
5
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7answers
317 views

Prove that if $n^2$ is divided by 3, then also $n$ can also be divided by 3.

$n\in \Bbb N$ Prove that if $n^2$ is divided by 3, then also n can also be divided by 3. I started solving this by induction, but I'm not sure that I'm going in the right direction, any ...
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3answers
92 views

Proof that if $3 \mid p^2$ then $3 \mid p$ [closed]

Course: Analysis (1st year course) Question: What does the formal proof of the following statement look like: if $3\mid p^2$ then $3\mid p$, with $p \in \Bbb Z$? Thank you. EDIT: I'd like to use ...
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4answers
31 views

Show that $\gcd(80,8a^2+1)=1$

Show that $\gcd(80,8a^2+1)=1$ Let $\gcd(80,8a^2+1)=d$, then we have: $d|80a^2+10,80a^2\Rightarrow\ d|10$ So $d=1\ or\ 2\ or\ 5\ or\ 10$ Obviously $d$ can't be $2\ or\ 10$,but how can we show $d$ can't ...