MIT6_042JS10_lec22_prob

MIT6_042JS10_lec22_prob - Problem 3 The following...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 31 Prof. Albert R. Meyer revised March 30, 2010, 1426 minutes In-Class Problems Week 8, Wed. Problem 1. (a) Use the Pulverizer to find the inverse of 13 modulo 23 in the range { 1 ,..., 22 } . (b) Use Fermat’s theorem to find the inverse of 13 modulo 23 in the range { 1 ,..., 22 } . Problem 2. (a) Why is a number written in decimal evenly divisible by 9 if and only if the sum of its digits is a multiple of 9? Hint: 10 1 (mod 9) . (b) Take a big number, such as 37273761261. Sum the digits, where every other one is negated: 3 + ( 7) + 2 + ( 7) + 3 + ( 7) + 6 + ( 1) + 2 + ( 6) + 1 = 11 Explain why the original number is a multiple of 11 if and only if this sum is a multiple of 11.
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Unformatted text preview: Problem 3. The following properties of equivalence mod n follow directly from its definition and simple prop-erties of divisibility. See if you can prove them without looking up the proofs in the text. (a) If a ≡ b (mod n ) , then ac ≡ bc (mod n ) . (b) If a ≡ b (mod n ) and b ≡ c (mod n ) , then a ≡ c (mod n ) . (c) If a ≡ b (mod n ) and c ≡ d (mod n ) , then ac ≡ bd (mod n ) . (d) rem( a,n ) ≡ a (mod n ) . Creative Commons 2010, Prof. Albert R. Meyer . MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT6_042JS10_lec22_prob - Problem 3 The following...

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