Let $S$ be a set of real numbers. We say that $S$ is strong if for any two distinct $a$ and $b$ from $S$, the number $a^2 + b\sqrt{2023}$ is rational. We say that $S$ is very strong if for every $a$ from $S$, the number $a\sqrt{2023}$ is rational. a) Prove that if $S$ is a very strong set, then it is also strong. b) Find the smallest natural number $k$ such that every strong set of $k$ distinct real numbers is very strong.
Problem
Source: XII International Festival of Young Mathematicians Sozopol 2023, Theme for 10-12 grade
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