soln8 - MATH 135 Algebra Solutions to Assignment 8 1 Solve...

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MATH 135 Algebra, Solutions to Assignment 8 1: Solve each of the following linear congruences. (a) 5 x 4 (mod 7) Solution: We make a table of values modulo 7. x 0 1 2 3 4 5 6 5 x 0 5 3 1 6 4 2 From the table we see that 5 x 4 (mod 7) when x 5 (mod 7). (b) 40 x 15 (mod 65) Solution: We have 40 x 15 (mod 65) when 40 x = 15 + 65 k for some integer k , or equivalently when 40 x + 65 y = 15 for some integer y . The Euclidean Algorithm gives 65 = 1 · 40 + 25 , 40 = 1 · 25 + 15 , 25 = 1 · 15 + 10 , 15 = 1 · 10 + 5 , 10 = 2 · 5 + 0 so we have gcd(40 , 65) = 5. Then Back-Substitution gives the sequence 1 , - 1 , , 2 - 3 , 5 so we have 40(5) + 65( - 3) = 5. Multiply both sides by 15 5 = 3 to get 40(15) + 65( - 9) = 15. Thus one solution to the given congruence is x = 15. Note that 65 5 = 13, so by the Linear Congruence Theorem, the general solution is x 15 2 (mod 13). Equivalently, x 2 , 15 , 28 , 41 or 54 (mod 65). (c) 391 x 119 (mod 1003) Solution: We have 391 x 119 (mod 1003) when 391 x + 1003 y = 119 for some integer y . The Euclidean Algorithm gives 1003 = 2 · 391 + 21 , 391 = 1 · 221 + 170 , 221 = 1 · 170 + 51 , 170 = 3 · 51 + 17 , 51 = 3 · 17 + 0 so gcd(391 , 1003) = 17. Back-Substitution then gives 1 , - 3 , 4 , - 7 , 18 so we have 391(18) + (1003)( - 7) = 17. Multiply both sides by 119 17 = 7 to get 391(126) + 1003( - 49) = 119. Thus one solution to the given congruence is x = 126. Note that 1003 17 = 59, so by the Linear Congruence Theorem, the general solution is x 126 8 (mod 59).
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2: (a) Find [12] - 1 in Z 29 . Solution: We must find x such that 12 x 1 (mod 29), that is 12 x + 29 y = 1 for some integer y . The Euclidean Algorithm gives 29 = 2 · 12 + 5 , 12 = 2 · 5 + 2 , 5 = 2 · 2 + 1 , 2 = 2 · 1 + 0 so gcd(12 , 29) = 1, and then Back-Substitution gives 1 , - 2 , 5 , - 12 so we have 12( - 12)+29(5) = 1. One solution to the congruence is x = - 12, so [12] - 1 = [ - 12] = [17] in Z 29 . (b) Solve [34] x = [18] in Z 46 . Solution: For x Z , to get 34 x 18 (mod 46), we need 34 x + 46 y = 18 for some integer y . The Euclidean Algorithm gives 46 = 1 · 34 + 12 , 34 = 2 · 12 + 10 , 12 = 1 · 10 + 2 , 10 = 5 · 2 + 0 so gcd(10 , 46) = 2, and then Back-Substitution then gives 1 , - 1 , 3 , - 4 so we have 34( - 4) + 46(3) = 2. Multiply both sides by 18 2 = 9 to get 34( - 36) + 46(27) = 18. Thus one solution to the congruence is x = - 36. Note that 46 2 = 23, so by the Linear Congruence Theorem, the general solution to the congruence is
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soln8 - MATH 135 Algebra Solutions to Assignment 8 1 Solve...

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