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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 11, please solve it briefly with every reason, its from subject Number theory.

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/Users/Dell/Downloads/SEM-6%20%20BOOKS/Elementary_Number Theory_7th_edition%20(1).pdf
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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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