IMO Shortlist 1978 problem 3
Dodao/la:
arhiva2. travnja 2012. Let
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be positive integers such that
![1 \le m < n](/media/m/b/a/f/baf6ea852325117cd5895cca0b9de519.png)
. In their decimal representations, the last three digits of
![1978^m](/media/m/2/1/b/21b8ff8fec1216b6c74f564847ebd048.png)
are equal, respectively, so the last three digits of
![1978^n](/media/m/5/e/2/5e2c3447ee4e4cf690a61ff22d2d4c02.png)
. Find
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
such that
![m + n](/media/m/f/a/3/fa3559150b25b783a0f1f7a70e27b04a.png)
has its least value.
%V0
Let $m$ and $n$ be positive integers such that $1 \le m < n$. In their decimal representations, the last three digits of $1978^m$ are equal, respectively, so the last three digits of $1978^n$. Find $m$ and $n$ such that $m + n$ has its least value.
Izvor: Međunarodna matematička olimpijada, shortlist 1978