contest math - Polynomial $P(a)=b,P(b)=c,P(c)=a$

Let $a,b,c$ be $3$ distinct integers, and let $P$ be a polynomial with integer coefficients.Show that in this case the conditions $$P(a)=b,P(b)=c,P(c)=a$$ cannot be satisfied simultaneously.

Any hint would be appreciated.

Answer

Hint: If $P(a)=b$ and $P(b)=c$ then $a-b$ divides $b-c$.