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Omni-MATH / There are 100 positive integers written on a board. At each step, Alex composes…

Problem

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problem

There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denominators to create 100 new numbers. Find the smallest positive integer nn{} such that regardless of the values of the initial 100 numbers, after nn{} steps Alex can arrange to have on the board only pairwise coprime numbers.
Plain-text mathematical notation (without MathML)
There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denominators to create 100 new numbers. 

Find the smallest positive integer n such that regardless of the values of the initial 100 numbers, after n steps Alex can arrange to have on the board only pairwise coprime numbers.
Original LaTeX notation
There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denominators to create 100 new numbers. 

Find the smallest positive integer $n{}$ such that regardless of the values of the initial 100 numbers, after $n{}$ steps Alex can arrange to have on the board only pairwise coprime numbers.

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Source and history

task_id
4bd414f1-97a0-52a0-9018-8763edbbbd32
task_key
test--4bd414f1-97a0-52a0-9018-8763edbbbd32
task_revision_id
1
config / split
/ test

Official source

initial import