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sum of reciprocals

Source: 2024 CTST P7

March 11, 2024
number theory

Problem Statement

For coprime positive integers a,ba,b,denote (a−1 mod b)(a^{-1}\bmod{b}) by the only integer 0≤m<b0\leq m<b such that am≡1(modb)am\equiv 1\pmod{b} (1)Prove that for pairwise coprime integers a,b,ca,b,c, 1<a<b<c1<a<b<c,we have(a−1 mod b)+(b−1 mod c)+(c−1 mod a)>a.(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})>\sqrt a. (2)Prove that for any positive integer MM,there exists pairwise coprime integers a,b,ca,b,c, M<a<b<cM<a<b<c such that (a−1 mod b)+(b−1 mod c)+(c−1 mod a)<100a.(a^{-1}\bmod{b})+(b^{-1}\bmod{c})+(c^{-1}\bmod{a})< 100\sqrt a.
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