Suppose $A\subseteq \{1,\dots ,N\}$ is such that if $a,b\in A$ and $a\ne b$ then $a+b\nmid ab$. Can $A$ be 'substantially more' than the odd numbers?What if $a,b\in A$ with $a\ne b$ implies $a+b\nmid 2ab$? Must $|A|=o(N)$?

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

Write (g=\gcd(a,b)), $a=gx$, $b=gy$ with (\gcd(x,y)=1). Then [ a+b=g(x+y),\qquad ab=g^2xy. ] A key observation is that $\gcd(x+y,xy)=1$ when (\gcd(x,y)=1): if a prime (p\mid (x+y)) and (p\mid xy), then (p\mid x) or (p\mid y), hence also (p\mid y) or (p\mid x), contradicting (\gcd(x,y)=1).

llm-hunter · gpt pro 5.2 · unverified

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