Difference between revisions of "Lifting the Exponent"

Latest revision as of 20:12, 19 June 2024

(Lemma from MAA official solution, 2020 AIME I Problems/Problem 12)

Denote $v_p(n)$ the highest power of prime $p$ that divides $n$ . Let $p$ be an odd prime, and let $a$ and $b$ be integers that are not multiples of $p$ such that $p \mid (a-b)$ . Let $n$ be a positive integer. Then $v_p(a^n - b^n) = v_p(a - b) + v_p(n)$ .

For more conclusions, see https://en.wikipedia.org/wiki/Lifting-the-exponent_lemma

edit by ~ab_godder

@@ Line 1: / Line 1: @@
-(From 2020 AIME I Problems/Problem 12)
+(Lemma from MAA official solution, 2020 AIME I Problems/Problem 12)
-Let <math>p</math> be an odd prime, and let <math>a</math> and <math>b</math> be integers relatively prime to <math>p</math> such that <math>p \mid (a-b)</math>. Let <math>n</math> be a positive integer. Then the number of factors of <math>p</math> that divide <math>a^n - b^n</math> is equal to the number of factors of <math>p</math> that divide <math>a-b</math> plus the number of factors of <math>p</math> that divide <math>n</math>.
+Denote <math>v_p(n)</math> the highest power of prime <math>p</math> that divides <math>n</math>.
+Let <math>p</math> be an odd prime, and let <math>a</math> and <math>b</math> be integers that are not multiples of <math>p</math> such that <math>p \mid (a-b)</math>. Let <math>n</math> be a positive integer. Then <math>v_p(a^n - b^n) = v_p(a - b) + v_p(n)</math>.
+For more conclusions, see https://en.wikipedia.org/wiki/Lifting-the-exponent_lemma
+edit by ~ab_godder

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Difference between revisions of "Lifting the Exponent"

Latest revision as of 20:12, 19 June 2024