IMO Shortlist 1993 problem N1
Dodao/la:
arhiva2. travnja 2012. A natural number
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
is said to have the property
![P,](/media/m/a/1/4/a14de1403fe9b971067bc4aac029b8a8.png)
if, for all
![a, n^2](/media/m/8/7/3/873c2168f305f7e95692c0094f80e7a0.png)
divides
![a^n - 1](/media/m/4/e/8/4e8a3d10b78bc1a39363a65938598454.png)
whenever
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
divides
a.) Show that every prime number
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
has property
b.) Show that there are infinitely many composite numbers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
that possess property
%V0
A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$
a.) Show that every prime number $n$ has property $P.$
b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$
Izvor: Međunarodna matematička olimpijada, shortlist 1993