Problem:Haoyu gives Mingze and Yihan two different positive integers, and tells them that the product of their numbers is less than $2022.$ They are also aware that they have two different positive integers. They have the following conversation: Mingze: I'm not sure whose number is larger. Yihan: Me neither. Mingze: Ah! Then my number is larger. Yihan: Now I know your number! What is Yihan's number?
$\textbf{(A)} ~41 \qquad \textbf{(B)} ~42 \qquad \textbf{(C)} ~43 \qquad \textbf{(D)} ~44 \qquad \textbf{(E)} ~45$
This question confuses me. Let's just say Mingze's number is $x$, while Yihan's number is $y$. I know that $xy<2022$, but then what? I don't know what to do with the conversation they had. Please enlighten me on how to solve this problem!
If Mingze's number were at least $45,$ he would know that Yihan's number must be smaller, or else the product would be at least $45\cdot 46 > 2022.$ Thus, his number is at most $44,$ and by the same reasoning, Yihan's is as well. In fact, Yihan's number is at most $43,$ since otherwise she would know that her number was larger. The only way that Mingze can then know that his number is larger is if it is either $43$ or $44.$ If Yihan's number were $42$ or smaller, she wouldn't be able to figure out which of $43$ or $44$ was Mingze's number, so her number must be $43,$ which is $\fbox{(C)}.$