N8

Part of VMEO III 2006 Shortlist

Problems(1)

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

For every positive integer

n

a_n/b_n

is the simplest form of the fraction

1+1/2+...+1/n

. Prove that for every pair of positive integers

(M, N)

we can always find a positive integer

m

(a_n, N) = 1

n = m, m + 1, ...,m + M

number theoryalgebra