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Determine the greatest integer $m$ for which $2102$ is $m$-additive

Source: Moldova TST 1993

August 8, 2023
number theory

Problem Statement

Positive integer qq{} is m−additivem-additive (m∈N,m≥2)(m\in\mathbb{N}, m\geq2) if there exist pairwise distinct positive integers a1,a2,…,ama_1,a_2,\ldots,a_m such that q=a1+a2+…+amq=a_1+a_2+\ldots+a_m and ai∣ai+1a_i | a_{i+1} for i=1,2,…,m−1i=1,2,\ldots,m-1. a) Prove that 19931993 is 88-additive, but 99-additive. b) Determine the greatest integer mm for which 21022102 is mm-additive.
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