Coprime numbers - number theory

Question

If there is a finite set of $k$ integers how can I prove that there is a coprime for each number in the set?

That is pretty obvious but what is the formal way to prove this?

Answer 1

Take any prime number strictly greater than every element in the set. (Such prime does exist because the set is finite and there are infinitely many prime numbers).

Answer 2

Take as this number their product + 1.

Answer 3

You are asking to prove a false thing, since you are implicitely assuming $0 \not \in \mathcal{S}$, where $\mathcal{S}:=\{a_1,\ldots,a_k\}\subseteq \mathbb{Z}$ for some $k \in \mathbb{N} \setminus \{0\}$ is your fixed set. Indeed, $\text{gcd}(0,z)=z$ for all $z \in \mathbb{Z}\setminus \{0\}$, so that $0\not\in \mathcal{S}$ is a necessary condition. But it's also sufficient: define the radical $$\text{rad}\colon \mathbb{Z} \setminus \{0\} \to \mathbb{N}\colon z \mapsto \prod_{p \in \mathbb{P}:\text{ }p\mid z^2}{p},$$ and then the (positive squarefree) integer $$A:=\text{rad}\left(\prod_{1\le i\le k}{|a_i|}\right)=\text{lcm}\left(\text{rad}(|a_1|),\ldots,\text{rad}(|a_k|)\right).$$ It's clear that for all $z \in \mathbb{Z}$ we have $$\text{gcd}(z,A)=1\text{ iff }\text{gcd}(z,a_i)=1 \text{ for all }i=1,\ldots,k$$ Defin $\mathcal{Z}$ the set of such integers. It's clear that $|\mathcal{Z}|=\infty$, for example by Chinese remainder theorem, or for example taking a particular classes of integers, like $kA\pm 1$ for $k \in \mathbb{Z}$. It's even possible to give asymptotic cardinality for $\mathcal{Z}$, indeed $$|\mathcal{Z} \cap [-x,x]| \sim 2x\frac{\varphi(A)}{A}\text{ with }x\to +\infty.$$ It shows a bit stronger fact: If $\omega(\text{rad}\left(\prod_{s \in \mathcal{S}}{s}\right))$ is finite (allowing even a not finite set $\mathcal{S}$, then there exists positive constants $A,B$ such that $$Ax<|\mathcal{Z}\cap [-x,x]|<Bx$$ for all positive integers $x$.

A more interesting exercise can be the fact that $\omega(A) \to \infty$ implies $\frac{\varphi(A)}{A} \to 0$.

Edit: as usual $\varphi:\mathbb{N}\setminus \{0\} \to \mathbb{N}:n \mapsto |\{m \in \mathbb{N}\setminus \{0\}:\text{gcd}(m,n)=1\}|$ denotes the Euler indicator, and $\omega: \mathbb{N}\setminus \{0,1\} \to \mathbb{N}:n \mapsto |\{p \in \mathbb{P}:p\mid n\}|$ represents the number of prime divisors of a integer $n\ge 2$..