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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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 ...
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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 .
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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
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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 (...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Count arrays with GCD as D

Given N ,I need to count the number of array of integers which satisfy the following conditions : ...
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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 ...
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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 ...
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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 ...
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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.
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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 $23^2 = 0$
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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 ...
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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 ...
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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 $b\mid u_1-u_2$. ...
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 $P(n+1)$: $$(n+1)!\ |\ 2^{n+1} ... 0answers 13 views Integer division and congruence exercise I'm just starting with integer division and congruence in an algebra course and I have this problem: Let a be an odd integer. Prove that \forall n \in \Bbb N:$$2^{n+2}\ |\ a^{2^n} - 1$$I've ... 0answers 19 views Linear factor divides a function I just came up with a simple question. If I have a polynomial function f(x_1,x_2,\ldots,x_n) and I know that when x_i=x_j, f=0. Then does it imply x_i-x_j divides f for all i\neq j? If yes, ... 1answer 25 views Help with congruence and divisibility exercise I'm starting to solve some problems of congruence and integer division, so the exercise is quite simple but I'm not sure I'm on the right track. I need to prove that the following is true for all n \... 2answers 57 views 7^{6} | (a+b+ab)^2 Find the value of a,b [closed] 7^{6} | (a+b+ab)^2 Find the value of a,b. I have used trial and error for a singular solution. But a generalized solution will be helpful. Provide me the concept to deal with this problem and ... 1answer 41 views Find all n such that n|1^n + 2^n + 3^n + \cdots + (n-1)^n where n \in \mathbb{Z}^+. Find all n such that$$n|1^n + 2^n + 3^n + \cdots + (n-1)^n$$where n \in \mathbb{Z}^+. I don't know how to start. n = 3, 5 are simple solutions. Induction seems strange since the divisor is ... 2answers 91 views cancelling out before evaluation of variable I'm been working on a theory, though my math is weak. Let's say I've managed to determine that I can arrive at an answer A by always using the formula BCD / D. Of ... 1answer 2k views Formula of MIPS (million instructions per second) Could you please help me to understand the mathematics behind MIPS rating formula? The performance of a CPU (processor) can be measured in MIPS. The formula for MIPS is:$$\text{MIPS} = \frac{\text{...
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 ...