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r(k) = r(k -1) for infinitely pos. integers k, sum of remainders n: 1,2...,n

Source: 1981 Hungary - Kürschák Competition p3

October 10, 2022
number theoryremainder

Problem Statement

For a positive integer nn, r(n)r(n) denote the sum of the remainders when nn is divided by 1,2,...,n1, 2,..., n respectively. Prove that r(k)=r(k−1)r(k) = r(k -1) for infinitely many positive integers kk.
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