Difference between revisions of "2008 OIM Problems/Problem 3"

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== Problem ==
 
== Problem ==
 
Let <math>m</math> and <math>n</math> be integers such that the polynomial <math>P(x) = x^3 + mx + n</math> has the following
 
Let <math>m</math> and <math>n</math> be integers such that the polynomial <math>P(x) = x^3 + mx + n</math> has the following
property: if <math>x</math> and <math>y</math> are integers and 107 divides <math>P(x)−P(y)</math>, then 107 divides <math>x-y</math>. Show that 107 divides <math>m</math>.
+
property: if <math>x</math> and <math>y</math> are integers and 107 divides <math>P(x)-P(y)</math>, then 107 divides <math>x-y</math>. Show that 107 divides <math>m</math>.
  
 
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
 
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Latest revision as of 16:31, 14 December 2023

Problem

Let $m$ and $n$ be integers such that the polynomial $P(x) = x^3 + mx + n$ has the following property: if $x$ and $y$ are integers and 107 divides $P(x)-P(y)$, then 107 divides $x-y$. Show that 107 divides $m$.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

OIM Problems and Solutions