Vrijeme: 08:11
Podijeljeni zajedno | Divided Together #1
U ovom lancu, za dva prirodna broja
i
, s
označavat ćemo najveći zajednički djelitelj od
i
. Neka su
prirodni brojevi takvi da je
. Odredi najmanju vrijednost koju može poprimiti
. Broj
je prost.
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
![(x, y)](/media/m/1/5/2/1520b43353795b60686f7df83802e90a.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
![a, b, c](/media/m/9/e/9/9e9dfe78930065fbe5a777e9b07c27c4.png)
![(a,b)+(b,c)+(c,a)=10^9+7](/media/m/c/1/e/c1e3dd5a679013ea177a34c1cbfb1ae0.png)
![a+b+c](/media/m/e/c/2/ec29939eba8a6bda191d0778cb4c458f.png)
![10^9+7](/media/m/6/7/4/674f5ca1807bf6b5f517e7043437d055.png)
In this chain, for positive integers
and
, we'll denote the greatest common divisor of
and
with
. Let
be positive integers such that
. Find the least possible value that
can obtain. The number
is prime.
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
![(x, y)](/media/m/1/5/2/1520b43353795b60686f7df83802e90a.png)
![a, b, c](/media/m/9/e/9/9e9dfe78930065fbe5a777e9b07c27c4.png)
![(a,b)+(b,c)+(c,a)=10^9+7](/media/m/c/1/e/c1e3dd5a679013ea177a34c1cbfb1ae0.png)
![a+b+c](/media/m/e/c/2/ec29939eba8a6bda191d0778cb4c458f.png)
![10^9+7](/media/m/6/7/4/674f5ca1807bf6b5f517e7043437d055.png)