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gcd and lcm

Source: 2015 Indonesia Math Olympiad Day 1 Problem 2

June 30, 2017

number theorygreatest common divisorleast common multiple

Problem Statement

For every natural number

a

and

b

, define the notation

[a,b]

as the least common multiple of

a

and

b

and the notation

(a,b)

as the greatest common divisor of

a

and

b

. Find all

n \in \mathbb{N}

that satisfies

4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)}