Lesson Math Olympiad level problem on divisibility numbers
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<H2>Math Olympiad level problem on divisibility numbers</H2> <H3>Problem 1</H3>Find the remainder when {{{1^2013}}} + {{{2^2013}}} + {{{3^2013}}} + . . . + {{{2012^2013)}}} is divided by 2013. <B>Solution</B> <pre> Let's group the addends in pairs {{{1^2013}}} + {{{2012^2013}}}, {{{2^2013}}} + {{{2011^2013}}}, {{{3^2013}}} + {{{2010^2013}}}, . . . . . . . . . . . . {{{1006^2013}}} + {{{1007^2013}}}. Thus, all the addends in the long sum are grouped in pairs this way. Now use the theorem (= the statement) that for all integers 'a' and 'b' and odd positive integer 'n' the sum {{{a^n}}} + {{{b^n}}} is a multiple of (a+b), so the sum {{{a^n}}} + {{{b^n}}} is divisible by (a+b) with zero remainder. It follows from well known polynomial decomposition {{{x^n}}} + {{{y^n}}} = {{{(x+y)*(x^(n-1) - x^(n-2)*y + x^(n-3)*y^2 - ellipsis - x*y^(n-2) + y^(n-1))}}}, which works for all odd integer 'n'. Now apply this theorem to each pair formed above. Since for each pair the sum of integers, that are the bases, is 2013, each and every pair is a multiple of 2013. Hence, the entire sum of these pairs is a multiple of 2013. It implies that long sum {{{1^2013}}} + {{{2^2013}}} + {{{3^2013}}} + . . . + {{{2012^2013)}}} is divisible by 2013, i.e. gives zero remainder when is divided by 2013. </pre> My other additional lessons on Miscellaneous word problems (section 3) in this site are - <A HREF=https://www.algebra.com/algebra/homework/word/misc/More-complicated-problems-on-finding-number-of-elements-in-finite-subsets.lesson>More complicated problems on finding number of elements in finite subsets</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-problems-by-the-Backward-method.lesson>Solving problems by the Backward method</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Minimax-problems-that-to-be-solved-MENTALLY-based-on-common-sense.lesson>Minimax linear problems to solve MENTALLY based on common sense</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-linear-optimization-problems-by-reduction-to-linear-function.lesson>Solving linear optimization problems without LP-method by reduction to linear function</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-one-special-linear-minimax-problem-in-100-D-space.lesson>Solving one special linear minimax problem in 100-D space by the Linear Programming method</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Miscellaneous-logical-problems.lesson>Miscellaneous logical problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Find-a-sequense-of-transformations-of-a-given-number-to-get-a-desired-number.lesson>Find a sequence of transformations of a given number to get a desired number</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/Upper-class-entertainment-Math-problems-for-all-ages.lesson>Upper class entertainment Math problems for all ages</A> - <A HREF=https://www.algebra.com/algebra/homework/word/misc/OVERVIEW-of-my-additional-lessons-on-Miscellaneous-word-problems-section-3.lesson>OVERVIEW of my additional lessons on Miscellaneous word problems, section 3</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I. Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
