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(n1, n2, n3) such that b1 b2 = b3

Source: 2009 Japan Mathematical Olympiad Finals, Problem 3

February 21, 2009
modular arithmeticnumber theory proposednumber theory

Problem Statement

Let k≥2 k\geq 2 be integer, n1, n2, n3 n_1,\ n_2,\ n_3 be positive integers and a1, a2, a3 a_1,\ a_2,\ a_3 be integers from 1,\ 2,\ \cdots ,\ k \minus{} 1. Let b_i \equal{} a_i\sum_{j \equal{} 0}^{n_i} k^{j}\ (i \equal{} 1,\ 2,\ 3). Find all possible pairs of integers (n1, n2, n3) (n_1,\ n_2,\ n_3) such that b_1b_2 \equal{} b_3.
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