Problem A. 415. (December 2006)
A. 415. Let a>b and n be arbitrary positive integers. Prove that n divides (an-bn).
(Proposed by Balázs Strenner, Budapest)
Deadline expired on January 15, 2007.
Solution. Let d=gcd(a,b), p=a/d and q=b/d. Since pn-qn divides an-bn, (pn-qn) divides (an-bn) and it is sufficient to prove n|(pn-qn). Therefore, it can be assumed that a and b are co-prime.
Note that a and b are relatively prime with an-bn as well.
Consider the multiplicative group of the reduced residue system modulo an-bn. Let c=ab-1. Since
the order of element c is exactly n.
By Lagrange's theorem, the order of element c divides the order of the group, therefore n divides (an-bn).
9 students sent a solution. 5 points: Gyenizse Gergő, Hujter Bálint, Kisfaludi-Bak Sándor, Kónya 495 Gábor, Korándi Dániel, Lovász László Miklós, Nagy 224 Csaba, Nagy 235 János, Tomon István.