MIT6_042JS10_lec21_prob

MIT6_042JS10_lec21_prob - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 29 Prof. Albert R. Meyer revised March 20, 2010, 738 minutes 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 Problem 2. (a) Use the Pulverizer to find integers x,y such that x 50 + y 21 = gcd(50 , 21) . · · (b) Now find integers x ,y with y > 0 such that x 50 + y 21 = gcd(50 , 21) · · Problem 3. For nonzero integers, a , b , prove the following properties of divisibility and GCD’S. (You may use the fact that gcd( a,b ) is an integer linear combination of a and b . You may not appeal to uniqueness of prime factorization because the properties below are needed to prove unique factorization.) (a) Every common divisor of
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MIT6_042JS10_lec21_prob - Massachusetts Institute of...

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