IMO 2015 Shortlist N3

Let m and n be positive integers such that m ą n. Define xk “ pmkq{pnkq for k “ 1,2,...,n1. Prove that if all the number...

imomathematicsolympiadshortlistnumber-theory

IMO 2015 Shortlist N3

Category: Number Theory

Problem

Let m and n be positive integers such that m ą n. Define xk “ pmkq{pnkq for k “ 1,2,...,n1. Prove that if all the numbers x1,x2,...,xn1 are integers, then x1x2 ¨¨¨xn`1 ´1 is divisible by an odd prime. (Austria)