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exists m with (a_n, N) = 1 for all n = m, m + 1, ...,m + M

Source: 2006 VMEO III Seniors 12.4 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463155_vmeo_iii

September 17, 2021
number theoryalgebra

Problem Statement

For every positive integer nn, the symbol an/bna_n/b_n is the simplest form of the fraction 1+1/2+...+1/n1+1/2+...+1/n. Prove that for every pair of positive integers (M,N)(M, N) we can always find a positive integer mm where (an,N)=1(a_n, N) = 1 for all n=m,m+1,...,m+Mn = m, m + 1, ...,m + M.
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