h7-sol - Solution of Homework 6: Basic Number Theory Q1....

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Solution of Homework 6 : Basic Number Theory Q1. Use the Euclidean algorithm to fnd (a) gcd(1529 , 14039), (b) gcd(1111 , 11111). Answer Note that in each iteration the larger value is replaced by the smaller value in the previous iteration and the smaller value is replaced by the remainder obtained in the previous iteration. This procedure is repeated until the remainder reaches 0 at which time we can conclude that the required gcd is the smaller value. (a) gcd(1529 , 14039): a b a mod b 14039 1529 278 1529 278 139 278 139 0 Thus, gcd(1529 , 14039) = 139 . (b) gcd(1111 , 11111): a b a mod b 11111 1111 1 1111 1 0 Thus, gcd(1111 , 11111) = 1 (that is, 11111 and 1111 are relatively prime).
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Q2. Compute 615 31 mod 713. Answer 31 = 2 4 + 2 3 + 2 2 + 2 1 + 2 0 = 16 + 8 + 4 + 2 + 1 615 31 mod 713 = (615 16 × 615 8 × 615 4 × 615 2 × 615) mod 713 = p (615 16 mod 713) × (615 8 mod 713) × (615 4 mod 713) × (615 2 mod 713) × (615 mod 713) P mod 713 . . . . . . (1) 615 mod 713
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This homework help was uploaded on 04/19/2008 for the course CS 182 taught by Professor W.szpankowski during the Fall '08 term at Purdue University-West Lafayette.

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h7-sol - Solution of Homework 6: Basic Number Theory Q1....

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