Can you do #6? 87Polynomial Congruences(c) x3 + 5x2 + 2x – 1 = 0 (mod 72)12. Solve the following polynomial congruences.(a) x? + 12x- 17 = 0 (mod 143)(b) r³ + 30x² + 27x + 23 = 0 (mod 45)3. Determine, without using a computer, all integers x such that the last threedigits of x° are the same as those of x.4. Let m = pi' P. Suppose the congruence f(x) = 0 (mod p) hasn; distinct solutions for 1 < i < k. Show that f(x) = 0 (mod m) hasnin2 · · · Nk distinct solutions.5. Let p be an odd prime. Show that x² = 1 (mod p*) has exactly two solu-tions for any k > 1.6. Let p be an odd prime. Show that x² = a (mod p*), k > 2 has exactly twosolutions if x² = a (mod p) has two solutions, and no solutions if x² = a(mod p) has no solutions.7. This exercise determines the solutions to x2 = 1 (mod 2k), k > 1.(a) Use a computer to determine the solutions for k =you see a general pattern?(b) Use the technique of Theorem 3.4.6 to construct the solutions to x² =1 (mod 16) from the set of solutions to æ² = 1 (mod 8). Whichsolutions of x² = 1 (mod 8) give solutions to x2 = 1 (mod 16)and why?= 1, 2, ..., 10. Do(c) Construct solutions to x? = 1 (mod 32) from those of x2 = 1(mod 16). Which solutions of x2 = 1 (mod 16) extend to solutionsof x2 = 1 (mod 32)? Do you see a pattern in these computationsthat will allow you to prove your guess from part (a)?(d) Determine all the solutions to x² = 1 (mod 2*), k > 5.8. How many solutions does the conaruenon

Name: Can you do #6? 87Polynomial Congruences(c) x3 + 5x2 + 2x – 1 = 0 (mod
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Description: Can you do #6? 87Polynomial Congruences(c) x3 + 5x2 + 2x – 1 = 0 (mod 72)12. Solve the following polynomial congruences.(a) x? + 12x- 17 = 0 (mod 143)(b) r³ + 30x² + 27x + 23 = 0 (mod 45)3. Determine, without using a computer, all integers x such tha

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Answered: 6. Let p be an odd prime. Show that a2…

Math

Advanced Math

6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod p) has two solutions, and no solutions if x² = a (mod p) has no solutions.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 92E

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