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Problem 1082

National olympiad, first round

Number theory Difficulty 6.1 Prove it

## Problem 1

Let $(\mathrm{a}, \mathrm{b}, \ldots, \mathrm{k})$ denote the greatest common divisor of the integers $\mathrm{a}, \mathrm{b}, \ldots \mathrm{k}$ and $[\mathrm{a}, \mathrm{b}, \ldots, \mathrm{k}]$ denote their least common multiple. Show that for any positive integers $\mathrm{a}, \mathrm{b}, \mathrm{c}$ we have $(\mathrm{a}, \mathrm{b}, \mathrm{c})^{2}[\mathrm{a}, \mathrm{b}][\mathrm{b}, \mathrm{c}][\mathrm{c}, \mathrm{a}]=[\mathrm{a}, \mathrm{b}$ , $c]^{2}(a, b)(b, c)(c, a)$ .

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

## Solution

If we express $\mathrm{a}, \mathrm{b}, \mathrm{c}$ as a product of primes then the $\mathrm{gcd}$ has each prime to the smallest power and the $1 \mathrm{~cm}$ has each prime to the largest power. So the equation given is equivalent to showing that $2 \min (\mathrm{r}, \mathrm{s}, \mathrm{t})+\max (\mathrm{r}, \mathrm{s})$ $+\max (\mathrm{s}, \mathrm{t})+\max (\mathrm{t}, \mathrm{r})=2 \max (\mathrm{r}, \mathrm{s}, \mathrm{t})+\min (\mathrm{r}, \mathrm{s})+\min (\mathrm{s}, \mathrm{t})+\min (\mathrm{t}, \mathrm{r})$ for non-negative integers $\mathrm{r}, \mathrm{s}, \mathrm{t}$ . Assume $\mathrm{r} \leq \mathrm{s} \leq \mathrm{t}$ . Then each side is $2 \mathrm{r}+\mathrm{s}+2 \mathrm{t}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.