IMO Shortlist 1974 problem 7
Dodao/la:
arhiva2. travnja 2012. Let
![a_i, b_i](/media/m/f/f/4/ff45f60bd65b0242630f87e505f2c9da.png)
be coprime positive integers for
![i = 1, 2, \ldots , k](/media/m/4/3/4/4344ac8427bc6d36f4504199f4e5c2da.png)
, and
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
the least common multiple of
![b_1, \ldots , b_k](/media/m/0/0/f/00fd681561b3b7f04ee1b0b37791eb98.png)
. Prove that the greatest common divisor of
![a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}](/media/m/8/f/3/8f32b6fcd8b691428c9f20c8d8832339.png)
equals the greatest common divisor of
%V0
Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$
Izvor: Međunarodna matematička olimpijada, shortlist 1974