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set of primes is infinite

Source: IMOC 2018 C4

August 16, 2021
number theorycombinatorics

Problem Statement

For a sequence {ai}i≥1\{a_i\}_{i\ge1} consisting of only positive integers, prove that if for all different positive integers ii and jj, we have ai∤aja_i\nmid a_j, then {p∣p is a prime and p∣ai for some i}\{p\mid p\text{ is a prime and }p\mid a_i\text{ for some }i\}is a infinite set.
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