coprime's questions - Chinese 1answer

45 coprime questions.

About the notation: Denotes $(a,b)$ the greatest common divisor between $a$ and $b$. Now, about the exercise, what I did was the following: Since $(m, n) = (s, n) = 1$, $ma + nb = sc + nd = 1$ then,...

In this section of the Wikipedia article on coprime integers, it is stated that: More generally, the probability of $k$ randomly chosen integers being coprime is $1/\zeta(k)$. where $\zeta$ is the ...

I am just getting started with some basic number theory and I was wondering: given two coprime natural numbers $a$ and $b$, is it true that $a+b$ is a prime number? My intuition says yes, because two ...

The following system is given $X \equiv a_1$ $mod$ $m_1$ $X \equiv a_2$ $mod$ $m_2$ such that $m_1, m_2 \in \mathbb{N} _{>1}$ and $m_1, m_2$ are not coprime. For which $a_1, a_2 \in \mathbb{Z} $...

let $n$ be an integer and $\phi$ is The Euler's Totient function. I want to know the probability for which $$\gcd ((\phi(n),n-1)=1 ) $$ I only know if $n$ is a prime number then the probability is 0 ...

Question: Suppose that $n = p_1p_2 \cdots p_k$, where $p_1, p_2, \ldots , p_k$ are distinct odd primes. Show that $a^{φ(n)+1} ≡ a\pmod n$ So I assume since n contains a bunch of distinct odd primes, ...

I'm stuck on the following question: For $a, b \in \Bbb Z$, assume that $ax + by = 4$ and $as + bt = 7$ for $x, y, s, t \in \Bbb Z$. Show that then $a$ and $b$ are relative prime. The following ...

Let n be odd. Write $n-1=2^{e}k$. Let $a\in\{1,...,n-1\}$ be a Miller-Rabin nonwitness: that is, $a^{k}\equiv1\,(mod\,n) $ or $a^{2^{i}k}\equiv-1\,(mod\,n)$ for some $i\in\{0,...,e-1\}$. Can I ...

Given an example with $(a,b)=1$ where $a=ux$ and $b=vy$ (with all variables being integers), obviously $(ux,vy) = 1$ directly, but does $(u,v) = 1$ as well? I am pretty sure it should but I am unsure ...

My simple problem relates with my question about sign-alternating sums of numbers, which opposite to coprime with $m$. $$1-\frac{1}{3}+\frac{1}{7}-\frac{1}{9}+\frac{1}{11}-\frac{1}{13}+\cdots=\frac{\...

For $x \in \mathbb{R}^+$, let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. Let $k \in \mathbb{N}$. Show that $2^{\{k \log_2(3)\}} < \dfrac{2}{1 + 2^{-k}}$ for $k > 1$. ...

Suppose $F_n$ is a free group of rank $n$, and $a$ is an element of $F_n$, such that $\exists b \in F_n(b^p=a)$ and $\exists c \in F_n(c^q=a)$, where $p$ and $q$ are coprime integers. Does there ...

Firstly: $[a]_m$ means $a \pmod m$ $m$ is co-prime with $n$, $\gcd(m,n)=1$ $[a_1]_{m}$ is coprime with $m$, $\gcd(a_1,m)=1$ $[a_2]_{n}$ is coprime with $n$, $\gcd(a_2,n)=1$ I find $x$ so that $[x]...

Let $m, \, n$ be coprime integers. (a) Let $G$ be an abelian group containing elements of orders $m$ and $n$. Prove that $G$ contains an element of order $mn$. (b) Deduce from part (a) that the ...

First equation is very popular - there are only odd numbers. Other words, numbers, which are coprime with $2$. $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{7}+\cdots=\frac{\pi}{4}$$ ...

What is the exact definition of coprime? I know that it basically means that there are no common factors, and one is not a multiple of the other. However, i recently learned that 1 is coprime to any ...

I know that s = 7 and t = -5, but I'm having trouble showing that they are not unique. I've been just guess and checking, is there a better way to compute this?

As one might know, the Stern Brocot tree elegantly as well as compactly models all rational numbers. I am now left wondering if a process like this tree modeling could be done not only for pairs but ...

Let given an operator $T$ with minimum polynomial $m(x)=p(x)q(x)$ such that $\gcd(p(x),q(x))=1$, then $\ker(p(T))= \text{Im}(q(T))$ So far I got $\exists r(x),s(x)$ such that $r(x)p(x)+s(x)q(x)=1$ ...

I would like to know if there is a nice way to show that the following two polynomials are coprime: $$ f_{n-1}(x)=\sum_{q=0}^{n-1}\sum_{m=0}^{q}{n-1 \choose 2q+1}{q \choose m}x^{n-2-2m}(-4)^m\\ f_{n-...

For integers $L$ and $M$ greater than $1,$ prove that the following sets are equivalent if and only if $L$ and $M$ are coprime. $$\bigg\{\large e^{\big(\tfrac{-i\text{ }2\pi \text{ }k}{L}\big)}\bigg\...

Is there a simple proof of the following statement? For any $a \in N$ exists such $b \in N$ that $a \perp b$ and $a + b^2$ is a composite number.

For an integer n, how many pairs (a, b) [suppose a is smaller than b] of coprime divisors of n exist such that (b-a) is divisible by 3 ? Advanced version of this question: Let F(n) denote the number ...

Does a prime labeling exist for all caterpillars, which are trees with every vertex being at most distance 1 from a central path? By a prime labeling, we mean a way to label the n vertices with the ...

I noticed that $\gcd(a^x,b^y)=\gcd(a,b)^z$ for a, b, x, y and z positive integers, My question is how to prove that? Additionally assuming the above is correct, when a and b are relatively prime (co-...

I observed that if $n$ has a primitive root then $c_1 \cdot c_2 ... \cdot c_{\phi(n)} \equiv -1 \ mod \ n$ otherwise, $c_1 \cdot c_2 ... \cdot c_{\phi(n)} \equiv 1 \ mod \ n$ where $c_i$'s are ...

In my foundations of math class we have just finished our section on number theory. I am having a really hard time with the questions involving co-primality, gcd's, and Bezout's identity. The ...

a+b+c=20 If a,b,c are coprime natural numbers to each other find number of triplets (a,b,c) ? Apart from manual counting what formula can we derive for such problem ?

Let $a$, $b$ and $c$ be non-zero integers. Suppose $a$ and $c$ are coprime. And suppose $b$ and $c$ are coprime. How can I then show that $ab$ and $c$ are coprime? From what I know so far this means ...

Would you please help me solve Exercise 2, which I repeat here: Suppose that $p$ and $q$ are relatively-prime positive integers. Show that if $\cos p \alpha$ and $\cos q \alpha$ are rational, then ...

Any even number $2n$ can be written as the sum of two primes, $p_{a}$ and $p_{b}$. For $n \geq 2$ this is the Goldbach Conjecture. $$ p_{a} + p_{b} = 2n $$ Why are $p_a$ and $2n$ co-prime? That is, $...

We are all familiar with the Mersenne primes $$M_n = 2^n-1$$ and we indeed know that there are some $M_n$ that are prime. However, it is still open whether there are infinitly many $M_n$ that are ...

I have an arbitrary number $x$. I would like to compute a number that is coprime to $x$ that's close(ish) to the square root of $x$. I don't need to find them all, and factoring $x$ is expensive. I ...

So, out of curiosity, I was wondering why $x (1 \to b)$ where $a$ & $b$ are co prime, why is $ax \bmod b$ distinct. For example, let $a = 5$, and $b= 8$: $\begin{array}{c|c|c} x & 5x & ...

If I have Large Number, and I want to find all possible CoPrime numbers for it, including Large Numbers above it, inside a certain set...how do I do that? I can only seem to find ways to find ...

The problem is to find all numbers $n$ such that all numbers $k>1$ smaller than $n$ and coprime with $n$ are prime.

Let $P(a, b)$, be a polynomial with monomials all degree $1$ or higher and integer coefficients (in other words, there are no constant terms). Prove that if $P(a, b) = 1$ (when replacing $a$ and $b$ ...

Say I have two rational numbers $a/b$ and $c/d$ where $a,b,c,d$ are integers and $a<b$ and $c<d$, and $a$ coprime with $b$, and $c$ coprime with $d$. Assume $b,d$ are free and not necessarily ...

Say I have two rational numbers $a/b$ and $c/d$ where $a,b,c,d$ are integers and $a<b$ and $c<d$, and $a$ coprime with $b$, and $c$ coprime with $d$. Assume $b,d$ are free and not necessarily ...

Is there a good algorithm for counting the numbers $x$ between $A$ and $B$ with $x$ and $N$ coprime? This is just like this question except for the range. The factorization of $N$ is known. I ...

Choose integers $q \ge 1$, $1 \le z_1,z_2 \le q$, and let $x_1,x_2$ be randomly chosen positive integers with the restriction that $x_1 \equiv z_1 \pmod q$ and $x_2 \equiv z_2 \pmod q$. What is the ...

Let $m$ and $n$ be two integers such that $m>n$. Then find the number of integers less than $m$ and relatively prime to $n$. I had come across a problem of this type with specific values for $m$ ...

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