MIT6_042JS10_lec21_sol

MIT6_042JS10_lec21_sol - Massachusetts Institute of...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 5

MIT6_042JS10_lec21_sol - Massachusetts Institute of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online