# coprime's questions - Chinese 1answer

45 coprime questions.

### 1 Prove that $(m,n) = (s,n) = 1$ if and only if $(ms,n)=1$

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,...

### 3 Probability of k random integers being coprimes

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 ...

### Is the sum of two coprime natural numbers prime?

3 answers, 771 views elementary-number-theory coprime
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 ...

### Abstract solving congruence system when modules are not coprime

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 $a_1, a_2 \in \mathbb{Z}$...

### 1 What is the probability for which :$\gcd ((\phi(n),n-1)=1 )$ for integers?

1 answers, 54 views probability totient-function coprime
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 ...

### Suppose that $n = p_1p_2 \cdots p_k$, where $p_1, p_2, .\ldots, p_k$ are distinct odd primes. Show that $a^{φ(n)+1} ≡ a\pmod n$

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, ...

### If $ax + by =$ prime, are then $a$ and $b$ relative prime?

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 ...

### Miller-Rabin nonwitnesses are relatively prime to n

0 answers, 18 views prime-numbers primality-test coprime
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 ...

### 1 The coprimality of 2 integers that are divisors of 2 larger coprimes?

1 answers, 11 views elementary-number-theory coprime
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 ...

### Problem with discrepancy of results, one of which is close to $\sqrt{\frac{\varphi^5}{5}}$

2 answers, 39 views prime-numbers riemann-zeta pi coprime

### 3 Finding a coprime of a general magnitude.

1 answers, 50 views prime-numbers coprime
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 ...