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I have been preparing for math olympiad recently and I came across this question.

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and I looked up for solution and it said enter image description here

Which I do not understand how the power of unknown such as 2013^n - 1803^n can be represented as (2013-1803)u when u is a polynomial. What is u in this equation and what kind of polynomial is it? Also how does this work?

Bill Dubuque
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Junsoo
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    This would refer to the identity $a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b +a^{n-3}b^2 +\cdots+b^{n-1})$, for $a,b\in\mathbb R$ and $n \in \mathbb N^*$. See this post for a proof. – DominikS Nov 29 '23 at 07:27
  • $u$ and $v$ are not polynomials; they're just integers. You can factor out $2013-1803$ from $2013^n-1803^n$ for the same reason $x-y$ divides $x^n-y^n$ – pancini Nov 29 '23 at 07:32
  • See here in the linked dupe for a simple way to solve such problems (exploiting the innate symmetry). See the "Linked" questions in the first dupe for further examples. The quoted solution is invoking the Factor Theorem $,a-b\mid a^n - b^n.\ \ $ – Bill Dubuque Nov 29 '23 at 07:41

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