Let $a, b,c$ be distinct positive integers, and suppose that $p = ab+bc+ca$ is a prime number. $(a)$ Show that $a^2,b^,c^2$ give distinct remainders after division by $p$. (b) Show that $a^3,b^3,c^3$ give distinct remainders after division by $p$.
Source: Netherlands TST for IMO 2017,day 2 problem 1
Tags: number theory
Let $a, b,c$ be distinct positive integers, and suppose that $p = ab+bc+ca$ is a prime number. $(a)$ Show that $a^2,b^,c^2$ give distinct remainders after division by $p$. (b) Show that $a^3,b^3,c^3$ give distinct remainders after division by $p$.