Question 26
Let $a$, $b$ and $c$ be positive integers, all distinct, and suppose that $p = a b + b c + c a$ is a prime number.
a) Prove that $a^{2}$, $b^{2}$ and $c^{2}$ leave different remainders upon division by $p$.
b) Prove that $a^{3}$, $b^{3}$ and $c^{3}$ leave different remainders upon division by $p$.
a) Prove that $a^{2}$, $b^{2}$ and $c^{2}$ leave different remainders upon division by $p$.
b) Prove that $a^{3}$, $b^{3}$ and $c^{3}$ leave different remainders upon division by $p$.