MIT6_042JS10_lec22_sol

MIT6_042JS10_lec22_sol - Massachusetts Institute of...

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6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 31 Prof. Albert R. Meyer revised March 30, 2010, 1434 minutes Solutions to 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 } . Solution. We first use the Pulverizer to find s,t such that gcd(23 , 13) = s 23 + t 13 , namely, · · 1 = 4 23 7 13 . · · This implies that 7 is an inverse of 13 modulo 23. Here is the Pulverizer calculation: x y rem( x,y ) = x q y · 23 13 10 = 23 13 13 10 3 = 13 10 = 13 (23 13) = ( 1) 23 + 2 13 · · 10 3 1 = 10 3 3 · = (23 13) 3 (( 1) 23 + 2 13)) · · · = 4 23 7 13 · · 3 1 0 = To get an inverse in the specified range, simply find rem( 7 , 23) , namely 16 . (b) Use Fermat’s theorem to find the inverse of 13 modulo 23 in the range { 1 ,..., 22 } . Solution. Since 23 is prime, Fermat’s theorem implies 13 23 2 13 1 (mod 23) and so rem(13 23 2 , 23) · is the inverse of 13 in the range
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MIT6_042JS10_lec22_sol - Massachusetts Institute of...

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