Given $A\subseteq \mathbb{N}$ let $M_A=\{n\ge 1:a\mid n\text{ for some }a\in A\}$ be the set of multiples of $A$. Find a necessary and sufficient condition on $A$ for $M_A$ to have density $1$.

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (A\subseteq\mathbb N) and list its elements in increasing order [ A={a_1<a_2<a_3<\cdots}. ] For each (k\ge 1) let (A_k={a_1,\dots,a_k}) and consider the finite set of multiples [ M_{A_k}={n\ge 1:\ a_i\mid n\text{ for some }i\le k}. ] Because (A_k) is finite, (M_{A_k}) is a finite union of arithmetic progressions, h…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.