2022 SSMO Team Round Problems/Problem 6

Problem

Let $n$ be a positive integer, and let $x$ be some variable. Define $P_{x,n}$ as the maximum fraction of elements in the set of the first $x$ natural numbers that may be contained in a subset $S$ such that if $k$ is an element of $S$ , then $nk$ is not. For example, $P_{3, 3}=\frac{2}{3}$ , since we take the set $\{1, 2\}$ . As $x$ approaches infinity, $P_{x,n}$ approaches a value $P_n$ . Given that ${\prod_{n=2}^{100}P_n}=\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers, find $a+b.$

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2022 SSMO Team Round Problems/Problem 6

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