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Number of carries

Source: Baltic Way 2017 Problem 19

November 11, 2017
number theory

Problem Statement

For an integer n≥1n\geq 1 let a(n)a(n) denote the total number of carries which arise when adding 20172017 and n⋅2017n\cdot 2017. The first few values are given by a(1)=1a(1)=1, a(2)=1a(2)=1, a(3)=0a(3)=0, which can be seen from the following: \begin{align*} 001 &&001 && 000 \\ 2017 &&4034 &&6051 \\ +2017 &&+2017 &&+2017\\ =4034 &&=6051 &&=8068\\ \end{align*} Prove that a(1)+a(2)+...+a(102017−1)=10⋅102017−19.a(1)+a(2)+...+a(10^{2017}-1)=10\cdot\frac{10^{2017}-1}{9}.
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