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Omni-MATH / S is a non-empty subset of the set {1,2,⋯,108}, satisfying:
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problem
is a non-empty subset of the set , satisfying:
(1) For any two numbers ( may not distinct), there exists , such that
$ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.
Find the largest possible value of .Plain-text mathematical notation (without MathML)
S is a non-empty subset of the set {1,2,⋯,108}, satisfying:
(1) For any two numbers a,b∈S ( may not distinct), there exists c∈S, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers a,b∈S ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.
Find the largest possible value of |S|.Original LaTeX notation
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:
(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.
Find the largest possible value of $ |S|$.Discussion
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