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6), then the congruence has a unique solution modulo p. 11. Show that the congruence x³ = 3 (mod 19) has no solutions, whereas x3 = 11 (mod 19) has three incongruent solutions.

6), then the congruence has a unique solution modulo p. 11. Show that the congruence x³ = 3 (mod 19) has no solutions, whereas x3 = 11 (mod 19) has three incongruent solutions.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 27E
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Question 11, please solve it briefly with every reason, its from subject Number theory. 

Fo X *Stephen H. Friedberg, Arnald CBCS BSc(Hons)Maths (3).pdf Elementary_Number Theory 7th x PDE 台 /Users/Dell/Downloads/SEM-6%20%20BOOKS/Elementary_Number Theory_7th_edition%20(1).pdf News Translate (D A TI 日一寸 has a solution for b = 2, 5, and 6. (b) Determine the integers a(1<a<p - 1) such that the congruence x = a (mod p) has a solution for p = 7, 11, and 13. 9. Employ the corollary to Theorem 8.12 to establish that if p is an odd prime, then (a) x² (b) x = -1 (mod p) is solvable if and only if p = 1 (mod 8). 10. Given the congruence x' = a (mod p), where p 5 is a prime and gcd(a, p) = 1, prove the following: (a) If p = 1 (mod 6), then the congruence has either no solutions or three incongruent solutions modulo p. (b) If p = 5 (mod 6), then the congruence has a unique solution modulo p. 11. Show that the congruence x= 3 (mod 19) has no solutions, whereas x = 11 (mod 19) has three incongruent solutions. 2. -1(mod p) is solvable if and only if p = 1 (mod 4). 168 ELEMENTARY NUMBLR THEORY
Transcribed Image Text:Fo X *Stephen H. Friedberg, Arnald CBCS BSc(Hons)Maths (3).pdf Elementary_Number Theory 7th x PDE 台 /Users/Dell/Downloads/SEM-6%20%20BOOKS/Elementary_Number Theory_7th_edition%20(1).pdf News Translate (D A TI 日一寸 has a solution for b = 2, 5, and 6. (b) Determine the integers a(1<a<p - 1) such that the congruence x = a (mod p) has a solution for p = 7, 11, and 13. 9. Employ the corollary to Theorem 8.12 to establish that if p is an odd prime, then (a) x² (b) x = -1 (mod p) is solvable if and only if p = 1 (mod 8). 10. Given the congruence x' = a (mod p), where p 5 is a prime and gcd(a, p) = 1, prove the following: (a) If p = 1 (mod 6), then the congruence has either no solutions or three incongruent solutions modulo p. (b) If p = 5 (mod 6), then the congruence has a unique solution modulo p. 11. Show that the congruence x= 3 (mod 19) has no solutions, whereas x = 11 (mod 19) has three incongruent solutions. 2. -1(mod p) is solvable if and only if p = 1 (mod 4). 168 ELEMENTARY NUMBLR THEORY
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