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MMO 124 Moscow MO 1947 one relative prime in 4 of 5 consecutive

Source:

July 20, 2019
number theoryrelatively primeconsecutivecombinatorics

Problem Statement

a) Prove that of 55 consecutive positive integers one that is relatively prime with the other 44 can always be selected.
b) Prove that of 1010 consecutive positive integers one that is relatively prime with the other 99 can always be selected.