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Let

\mathbb{P}

be the set of all primes and let

M

be a subset of

\mathbb{P}

with at least three elements. Suppose that for all

k \geq 1

and for all subsets

A=\{p_1,p_2,\dots ,p_k \}

M

A\neq M

, all prime factors of

p_1p_2\dots p_k-1

are in

M

. Prove that

M=\mathbb{P}

2