MIT6_042JS10_lec21_sol

MIT6_042JS10_lec21_sol - Massachusetts Institute of...

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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 29 Prof. Albert R. Meyer revised March 29, 2010, 739 minutes Solutions to In-Class Problems Week 8, Mon. Problem 1. A number is perfect if it is equal to the sum of its positive divisors, other than itself. For example, 6 is perfect, because 6 = 1 + 2 + 3 . Similarly, 28 is perfect, because 28 = 1 + 2 + 4 + 7 + 14 . Explain why 2 k − 1 (2 k − 1) is perfect when 2 k − 1 is prime. 1 Solution. If 2 k − 1 is prime, then the only divisors of 2 k − 1 (2 k − 1) are: 1 , 2 , 4 , ..., 2 k − 1 , (1) and 1 (2 k − 1) , 2 (2 k − 1) , 4 (2 k − 1) , ..., 2 k − 2 (2 k − 1) . (2) · · · · The sequence ( 1 ) sums to 2 k − 1 (using the formula for a geometric series, 2 and likewise the se- quence ( 2 ) sums to (2 k − 1 − 1) (2 k − 1) . Adding these two sums gives 2 k − 1 (2 k − 1) , so the number · is perfect. Problem 2. (a) Use the Pulverizer to find integers x,y such that x 50 + y 21 = gcd(50 , 21) . · · Creative Commons 2010, Prof. Albert R. Meyer . 1 Euclid proved this 2300 years ago. About 250 years ago, Euler proved the converse: every even perfect number is of this form (for a simple proof see http://primes.utm.edu/notes/proofs/EvenPerfect.html ). As is typical in number theory, apparently simple results lie at the brink of the unknown. For example, it is not known if there are in number theory, apparently simple results lie at the brink of the unknown....
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This note was uploaded on 05/27/2011 for the course CS 6.042J taught by Professor Prof.albertr.meyer during the Spring '11 term at MIT.

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MIT6_042JS10_lec21_sol - Massachusetts Institute of...

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