# 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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### Prove or disprove f an integer is divisible by 4, then it is divisible by 8

I need to know if I can prove or disprove if an integer is divisible by 4 then it is divisible by 8,for this question should i just show a value like 12 to show this statement is wrong or what? How ...
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### In any set of ten consecutive positive integers, there is one that is coprime with each one of the others [duplicate]

Let $a$ be a postive integer and let $A=\{a,a+1,a+2,\ldots,a+9\}.$ Show that there exists some $i$ such that for any $j\neq i$ we have $(a+i,a+j)=1$
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### If a number cannot be…

If there exists such a number which cannot be divided by some other number, which is equivalent to, or smaller than the square root of itself, it is a prime number. This is a rather trivial theorem ...
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### Checking whether a number is prime or composite

This is a question that came up while I was doing an exercise. I ended up with the number $$200! + 1$$ and I want it to be composite but I don't know of any methods to check whether a number is ...
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### why is 6 divided by 1245 207.5? instead of 207 remainder 3?

Help, 6 / 1245 = 207.5? I did long division to get my answer. But when i calculate it myself i end up with 207 remainder 3 ,how does that translate into .5? I don't understand.
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### If $x$ divides $x-z$, then $x$ divides $z$

For any integer x and z , if $x|(x-z)$ then $x|z$ My attempt: suppose $x|(x-z),$ let $y= x-z$ $x|y$ means there is any integer r such that $y=r*x$ So $x-z=rx$, which equals $(x-z)/(x) =r$ ...
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### Why test of divisible by $12$ works with $3$ and $4$ but not with $2$ and $6 ?$

Test of divisible by $4 ,$ last two digit must be divisible by $4 ,$ since $100$ is always divisible by $4$ remaining two digit $,$ we need to check $.$ Test of divisible by $3 ,$ sum digits must be ...
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### What number from an integer range has the most divisors? [duplicate]

I've been wondering. What number from an integer range (-2 147 483 647/+2 147 483 647) has the most divisors and how many is that?
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### Divisibility of a series

I use the notation $123 \dots (z)$ to represent a number that looks like a concatenated string of consecutive integers up to $z\in \mathbb{N}$. E.g. $123 \dots (15)$ denotes $12346789101112131415$. I ...
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### Prove divisibility: if $a\mid (b-d)$ and $a\mid (c-e)$, then $a\mid (bc-de)$

I have this math question. It states: Show that for any $a , b ,c, d, e \in \mathbb{Z^+}$, if $a\mid (b-d)$ and $a\mid (c-e)$, then $a\mid (bc-de)$. I'm not 100% sure as to how to start this ...
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### Number Theory Homework: Find 3 consecutive integers…

I have this problem assigned for homework, and I'm a bit confused as to how to solve it: Obtain three consecutive integers, the first of which is divisible by a square, the second by a cube, and the ...
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### Prove that if 2 divides $x^2-5$ then 4 divides $x^2-5$

so I have to prove this and I use two different types of proof and I came to a contradicting result. Can someone point out an error I made? Using a direct proof: If 2 divides $x^2-5$ than $x^2-5=2k$ ...
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### Divisibility proof with co-prime numbers

Let $a,b$ be co-prime. Prove that for every integer $n > a$ it is true that $a | (n + kb)$ for some $k$ with $0 <= k < a$. I have a feeling this is very basic stuff and I feel like there is ...
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### if $5\nmid a$ or $5\nmid b$, then $5\nmid a^2-2b^2$.

I have a homework as follow: if $5\nmid a$ or $5\nmid b$, then $5\nmid a^2-2b^2$. Please help to prove it. EDIT: MY ATTEMPT Suppose that $5\mid a^2-2b^2$, then $a^2-2b^2=5n$,where $n\in Z$, then ...
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### Simple question about dividing by zero, $y=\frac{x}{x}$ when $x=0$

Is there a rule that says you have to simplify equations before evaluating them? Would $y=\frac{x}{x}$ at $x=0$ be $1$ or undefined, since without reducing it, you'd divide by $0$. I know the equation ...
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### Prove divisibility with gcd: If $ar+bs=d=\gcd(a,b)$, then $r$ and $s$ are relatively prime

I have this math problem. The question is: Suppose that $a$ and $b$ are positive integers, with $d = \gcd(a, b)$. Suppose that there exists integers $r$ and $s$ so that $ar + bs= d$. We ...
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### If a divisor of $pq-1$ divides the LCM of $p-1$ and $q-1$, then it also divides the GCD of these two numbers

Suppose that $p,q$ are distinct odd primes. Suppose an integer $k$ divides $pq-1$ and also $k|\operatorname{lcm}(p-1,q-1)$. Show that $k|\operatorname{gcd}(p-1,q-1)$. I've spent ages looking at ...
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### Prove that $AB\mid CD$

I have this math question that I'm kind of confused on. This is the question: Let $A, B, C$ and $D$ be integers with $A \mid C$ and $B \mid D$ show that $$AB \mid CD.$$ I'm not 100% sure ...
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### Proof that $n+k+3$ divides $n(n+1)(n+2)(n+3) - k(k+1)(k+2)(k+3)$.

I'm looking for proof that $$(n+k+3) \mid n(n+1)(n+2)(n+3) - k(k+1)(k+2)(k+3)\\ n,k \in \mathbb N^*, n>k$$ I tried using induction, but i'm not sure how it would work with 2 parameters.
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### LCM of $n$ consecutive natural numbers

Is there an efficient way to calculate the least common multiple of $n$ consecutive natural numbers? For example, suppose $a = 3$ and $b = 5$, and you need to find the LCM of $(3,4,5)$. Then the LCM ...
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### Why are the disadvantages of approximation?

When I do 1/3 = 0.33333 but when I do 3*0.33333 then answer is 0.99999, I mean not whole 1 but 0.1 less than 1. What are the drawbacks of this think/rule since it's very basic math. Also why One cant ...
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### Prove that $GCD(a,b)=1$ if for all natural numbers $c, a|bc$ then $a|c$.

I'm trying to prove a theorem out of my text: Theorem: Let $a$ and $b$ be natural numbers. Then $GCD(a,b)=1$ if and only if for all natural numbers $c$, if $a|bc$ then $a|c$. I did come across this ...
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### If $n \mid a^2$, what is the largest $m$ for which $m \mid a$?

Given $n$, what is the largest $m$ such that $m \mid a$ for all $a$ with $n \mid a^2$? This is a generalization of if $40|a^2$ prove that $20|a$ when $a$ is an integer where $n=40$ and $m=20$. Here ...
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### Is my limited understanding of division and gcd on track?

Hello I am trying to make sense of some beginner theorems and propositions in number theory. I am wanting to also know if what I am saying is valid or just completely wrong. I am wanting to show that ...
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### Palindromes and LCM

A palindrome is a number that is the same when read forwards and backwards, such as $43234$. What is the smallest five-digit palindrome that is divisible by $11$? I'm probably terrible at math but ...
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### Divisibility test using perhaps binomial thorem

I have to determine if $17^{21} + 19^{21}$ is divisible by any of the following numbers (a) 36 (b) 19 (c) 17 (d) 21. I am trying to find using binomial expansion but getting stuck up with one or two ...
Proving this can be done as follows: consider a finite group G and elements $g_i \in G$ for some integer $i$. Now consider $\langle g_i \rangle = \{g_i^n: n\geq 0\}$, a generator. It can be proved ...
### If $r$ is a nonzero solution $x^2 + ax + b$, prove that $r | b$
I know that if $r$ is a solution, then there exist two factors of $b$ that when multiplied equal $b$ and that $r$ is one of them. So clearly $r$ divides $b$, but I don't know if there is any other way ...