4023f11ps2a

4023f11ps2a - Homework #2 Solutions Due: September 6, 2011...

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Homework #2 Solutions Due: September 6, 2011 1. Find d = gcd( m; n ) and write d in the form d = sm + nt for each of the following pairs of integers m and n . (a) m = 56, n = 72. I Solution. Use the matrix method illustrated in class for the Euclidean algo- rithm. 1 0 56 0 1 72 ( ¡ 1) ¢ 56+72=16 ¡! 1 0 56 ¡ 1 1 16 ( ¡ 3) ¢ 16+56=8 ¡! 4 ¡ 3 8 ¡ 1 1 16 ( ¡ 2) ¢ 8+16=0 ¡! 4 ¡ 3 8 ¡ 9 7 0 Hence gcd(56 ; 72) = 8 and 4 ¢ 56 + ( ¡ 3) ¢ 72 = 8. J (b) m = 119, n = 272. I Solution. 1 0 119 0 1 271 ( ¡ 2) ¢ 119+272=34 ¡! 1 0 119 ¡ 2 1 34 ( ¡ 3) ¢ 34+119=17 ¡! 7 ¡ 3 17 ¡ 2 1 34 ( ¡ 2) ¢ 17+34=0 ¡! 7 ¡ 3 17 ¡ 16 7 0 Hence gcd(119 ; 272) = 17 and 7 ¢ 119 + ( ¡ 3) ¢ 272 = 17. J (c) m = 513, n = 187. I Solution. 1 0 513 0 1 187 ( ¡ 2) ¢ 187+513=139 ¡! 1 ¡ 2 139 0 1 187 ( ¡ 1) ¢ 139+187=48 ¡! 1 ¡ 2 139 ¡ 1 3 48 ( ¡ 2) ¢ 48+139=43 ¡! 3
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4023f11ps2a - Homework #2 Solutions Due: September 6, 2011...

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