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

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2
votes
2answers
42 views

A number is divisible by 13 [duplicate]

I am studying divisibility and come across this rule. I think the rule is too complicated and hard to understand and remember. What is the best way to judge whether a number is divisible by 13 without ...
2
votes
2answers
36 views

Limit of indeteminate form $1^{(∞)}$

If we consider the function $f(x)=[(ax+1)/(bx+2)]^{x}$ where $a$,$b$ >$0$ and a I tried as follows]1 But at end i got stuck .
-1
votes
1answer
10 views

Analogue Tape how long do I have to record?

If I have 1200ft (feet) of tape. How long will I be able to record for at 7.5ips (inches per second) Thank you
3
votes
4answers
113 views

Proof that $3 \mid \left( a^2+b^2 \right)$ iff $3 \mid \gcd \left( a,b\right)$

After a lot of messing around today I curiously observed that $a^2+b^2$ is only divisible by 3 when both $a$ and $b$ contain factors of 3. I am trying to prove it without using modular arithmetic (...
9
votes
8answers
523 views

Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.

For elements $a$ and $b$ in the ring $\Bbb{Z}$ prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$. I tried proving it but I just don't manage to. Maybe I am missing some basic claims in the ...
0
votes
2answers
462 views

How many 4-digit numbers with $3$, $4$, $6$ and $7$ are divisible by $44$?

Consider all four-digit numbers where each of the digits $3$, $4$, $6$ and $7$ occurs exactly once. How many of these numbers are divisible by $44$? My attack: There are $24$ possible four digit ...
-1
votes
2answers
27 views

If $a,b,c \in Z$, $\gcd(a-b,b-c) = \gcd(a-b,a-c)$ [closed]

I need to prove that for every three integers $(a,b,c)$, the $\gcd(a-b,b-c) = \gcd(a-b,a-c)$. Assuming that a $a \ne b$. Having: $d_1 = \gcd(a-b,b-c)$ $d_2 = \gcd(a-b,a-c)$ How do i prove $d_1 = ...
0
votes
1answer
37 views

determine odd number pattern?

How can I determine series of such numbers which when keep dividing by 2 always produce odd quotient? For example: 15 15/2 = 7 (odd) (take only integer(floor) part) 7/2 = 3 (again odd) 3/2 = 1 (again ...
0
votes
1answer
57 views

Prove that $n^2+11n+2$ is not divisible by $12769$ [duplicate]

My Attempt : Prime factorisation of $12769$ is $113^2$ $n^2+11n+2-113^2m=0$ The conjugate of this quadratic equation becomes: $\sqrt {113 (113m+1)} $ which can never be a rational as (113,113m+1)...
1
vote
5answers
123 views

Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer)

Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer) It's obvious that if we show $113$ doesn't divide $n^2+11n+2$ we are done...
45
votes
2answers
823 views

Fractals using just modulo operation

Let us calculate the remainder after division of $27$ by $10$. $27 \equiv 7 \pmod{10}$ We have $7$. So let's calculate the remainder after divison of $27$ by $7$. $ 27 \equiv 6 \pmod{7}$ Ok, so ...
2
votes
1answer
28 views

Find number of Distinct remainders when $2009$ is divided by all natural numbers

Find number of Distinct remainders when $2009$ is divided by all natural numbers. obviously if we divide $2009$ by numbers greater than $2009$ remainder is $2009$ so we have to find remainders when $...
1
vote
1answer
39 views

Ratio vs division

I remember reading somewhere that in ancient times they were not treating a ratio like a division as we do. I was wondering is there a subtle distinction between the concept of the ratio and the idea ...
0
votes
1answer
65 views

When and why does this divide?

I've been working a lot with forms of this type, $\lfloor\frac{f}{g}\rfloor-\lfloor\frac{f-1}{g}\rfloor=1$ if $g|f$ and $0$ otherwise. This is valid for any expression $f$ and $g$ of natural numbers ...
1
vote
4answers
81 views

A positive integer (in decimal notation) is divisible by 11 $ \iff $ …

(I am aware there are similar questions on the forum) What is the Question? A positive integer (in decimal notation) is divisible by $11$ if and only if the difference of the sum of the digits in ...
4
votes
4answers
88 views

If $p > 3$ is prime, then $12 $ divides $p^2 - 1$

First up, I know there are a lot of similar questions with 24, not 12. So bare with me please :) What is the Question? Consider the following numbers of the form $p^2 - 1$ where $p$ is prime. $$5^2 ...
1
vote
3answers
32 views

Find the remainder of $\sum_{i=0}^{99} 2^{i^2}$ when dividing by 7 and determine if the quotient is even or odd

I've recently had this problem in an exam and couldn't solve it. Find the remainder of the following sum when dividing by 7 and determine if the quotient is even or odd: $$\sum_{i=0}^{99} 2^{i^2}$$ ...
1
vote
1answer
38 views

Problem to find all $n$ in following situation [closed]

Find all $n>1$ such that $1^{n} + 2^{n} + 3^{n} +\cdots + (n-1)^{n}$ divisible by $n$. I'm not good at Number Theory so , give elementary answer.
0
votes
1answer
18 views

Total number of integral solutions to the factors of a given numbet

Let $a$ be a factor of $120$ then what are the total number of positive integral solutions to $xyz=a$ including 120. The answer is $320$ . After wasting almost $15$ mins in getting the factors of each ...
6
votes
4answers
9k views

The product of n consecutive integers is divisible by n! (without using the properties of binomial coefficients)

How can we prove, without using the properties of binomial coefficients, the product of n consecutive integers is divisible by n factorial?
0
votes
0answers
37 views

Prove that $\frac{a^2+b^2}{1+ab}$ must be a perfect square [duplicate]

if $a$ and $b$ are positive integers and if $1+ab$ divides $a^2+b^2$ then prove that the quotient must be a perfect square. Let $$\frac{a^2+b^2}{1+ab}=k$$ where $k$ is some positive integer now $$a^...
1
vote
0answers
360 views

Count arrays with GCD as D

Given N ,I need to count the number of array of integers which satisfy the following conditions : ...
2
votes
1answer
40 views

Dividing primes

Let $p_1,\dots , p_{n+1}$ be distinct primes, let $\alpha_1, \dots , \alpha_n$ be integers, and let $a,b$ be integers. Suppose we had the equation: $$b^2p_{n+1} = a^2p_1^{\alpha_1}\dots p_n^{\alpha_n}...
2
votes
2answers
63 views

Factorial Divisibility

Let $a$ and $b$ be positive integers greater than one. With that in mind, $$(a \cdot b)!$$ is not necessarily divisible by: a) $$a!^b$$ b) $$b!^a$$ c) $$a! \cdot b!$$ d) $${2}^{ab}$$ By brute-...
51
votes
10answers
13k 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 ...
7
votes
1answer
86 views

Show that among any consecutive $16$ natural numbers one is coprime to all others

Show that among any consecutive $16$ natural numbers one is coprime to all others. Is it useful to use the division algorithm on $16$? $16k,16k+1,16k+2,...16k+15$
12
votes
6answers
6k views

How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?

I can see that this works for any integer $n$, but I can't figure out why this works, or why the number $42$ has this property.
10
votes
2answers
131 views

Is a function of $\mathbb N$ known producing only prime numbers?

It is well known that a polynomial $$f(n)=a_0+a_1n+a_2n^2+\cdots+a_kn^k$$ is composite for some number $n$. What about the function $f(n)=a^n+b$ ? Do positive integers $a$ and $b$ exists such ...
1
vote
2answers
91 views

Any digit written $6k$ times forms a number divisible by $13$

Any digit written $6k$ times (like $111111$, $222222222222222222222222$, etc.) forms a number divisible by $13$. (source: a solution taken from careerbless) I tested with many numbers and it seems ...
3
votes
4answers
81 views

Proving divisibility for $256 \mid 7^{2n} + 208n - 1$

I can't come up with a way of proving this: $$256 \mid 7^{2n} + 208n - 1\\ \forall n \in \Bbb N$$ I've tried by induction but couldn't see when to apply the inductive hypothesis... $$P(n+1) = 7^{2n+...
1
vote
0answers
22 views

How Euclidian Algorithm for division works with algebric expressions?

I am attending an introductory Number Theory class for Computer Science focused on cryptography. I have done some exercises with integers number but I have two exercises in which appears algebric ...
2
votes
1answer
23 views

Finding quotient and remainder for a division

We are starting with division and congruence in my algebra course... this is one of the first exercises for the division algorithm. I've done the first that were given with fixed values but now I have ...
13
votes
3answers
1k views

How to solve this algorithmic math olympiad problem?

So, today we had a local contest in my state to find eligible people for the international math olympiad "IMO" ... I was stuck with this very interesting algorithmic problem: Let $n$ be a natural ...
1
vote
1answer
35 views

Find all $n \in \Bbb Z$ such that $n^2 + n + 1$ divide $n^3-22$

I need help with this problem: Find all $n \in \Bbb Z$ such that $n^2 + n + 1$ divide $n^3-22$. I've got to a point where I know that $n^2 + n + 1 | -21$. So it should be among {${-21, -7, -3, -1, 1,...
2
votes
1answer
81 views

Prove that $(a+b+c)^{333}-a^{333}-b^{333}-c^{333}$ is divisible by $(a+b+c)^{3}-a^{3}-b^{3}-c^{3}$

Prove that $$(a+b+c)^{333}-a^{333}-b^{333}-c^{333}$$ is divisible by $$(a+b+c)^{3}-a^{3}-b^{3}-c^{3},$$ where $a,,b,c -$ integers, such that $(a+b+c)^{3}-a^{3}-b^{3}-c^{3}\not =0$ My work ...
4
votes
2answers
282 views

Sum of squares of integers divisible by 3

Suppose that $n$ is a sum of squares of three integers divisible by $3$. Prove that it is also a sum of squares of three integers not divisible by $3$. From the condition, $n=(3a)^2+(3b)^2+(3c)^2=9(a^...
0
votes
2answers
38 views

Demonstrate that $\int_0^1{\frac{(x^2+x+1)^{4n+1}- x}{x^2+1}dx}$ is a rational number

I thought about proving $x^2+1$ divides $(x^2+x+1)^{4n+1}- x$ , but I don't know how.
0
votes
1answer
41 views

Number of solutions for $n^5 + 2 n^4 + n^3 - 3n + 2 $ mod $ 23^2 = 0$, where $0 \leq n < 23^2$ and $n\in \mathbb{N}$

$0 \leq n < 23^2$ and $n\in \mathbb{N}$ For how many $n$ $n^5 + 2 n^4 + n^3 - 3n + 2 $ mod $ 23^2 = 0$
3
votes
2answers
58 views

if $p\mid a$ and $p\mid b$ then $p\mid \gcd(a,b)$

I would like to prove the following property : $$\forall (p,a,b)\in\mathbb{Z}^{3} \quad p\mid a \mbox{ and } p\mid b \implies p\mid \gcd(a,b)$$ Knowing that : Definition Given two natural ...
7
votes
0answers
79 views

Which prime factors of $8^{8^8}+1$ are known?

We have the partial factorization $$8^{8^8}+1=(2^{2^{24}}+1)\cdot (2^{2^{25}}-2^{2^{24}}+1)$$ The first factor is $F_{24}$. It is composite, but no prime factor is known. A prime factor of the ...
1
vote
3answers
55 views

Integers divide several solutions to Greatest Common Divisor equation

I'm not sure about the topic's correctness but my problem is following: Suppose $u_1,v_1$ and $u_2,v_2$ are two different solutions for $au_i + bv_i = 1$, then $a \mid v_2-v_1$ and $b\mid u_1-u_2$. ...
0
votes
0answers
13 views

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} ...
1
vote
0answers
13 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 ...
0
votes
0answers
19 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, ...
0
votes
1answer
25 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
votes
2answers
57 views

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

$7^{6} | (a+b+ab)^2$ 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
votes
1answer
41 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 is ...
2
votes
2answers
91 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 ...
1
vote
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} = \frac{\text{...
0
votes
1answer
17 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 ...