# [SOLVED]Polynomial with integer coefficients

#### anemone

##### MHB POTW Director
Staff member
Let $a,\,b,\,c$ be three distinct integers and $P$ 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.

##### Well-known member
The polynomial is P(x)

we have m-n divides P(m) - p(n)

Let the given condition is true

So $a-b | P(a) - p(b)$

or $a-b | b- c$

Similarly

$b - c | c- a$

And $c- a | a - b$

From above 3 have

$a-b | b-c | c- a | a-b$

So all are same and hence a = b= c which is a contradiction.

so condition can not be satisfied simultaneously