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1.17. Find all values of x between 0 and m - 1 that are solutions of the following congruences. (Hint. If you can't figure out a clever way to find the solution(s), you can just substitute each value x = 1, x = 2,..., x = m - 1 and see which ones work.) (a) x + 17 = 23 (mod 37). (b) x + 42 = 19 (mod 51). (c) x² = 3 (mod 11).

1.17. Find all values of x between 0 and m - 1 that are solutions of the following congruences. (Hint. If you can't figure out a clever way to find the solution(s), you can just substitute each value x = 1, x = 2,..., x = m - 1 and see which ones work.) (a) x + 17 = 23 (mod 37). (b) x + 42 = 19 (mod 51). (c) x² = 3 (mod 11).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.6: Inequalities
Problem 80E
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if possible a,b,c,d,e,f,g

### 1.17. Find all values of \( x \) between 0 and \( m - 1 \) that are solutions of the following congruences. **Hint**: If you can’t figure out a clever way to find the solution(s), you can just substitute each value \( x = 1, x = 2, \ldots, x = m - 1 \) and see which ones work. (a) \( x + 17 \equiv 23 \pmod{37} \). (b) \( x + 42 \equiv 19 \pmod{51} \). (c) \( x^2 \equiv 3 \pmod{11} \). (d) \( x^2 \equiv 2 \pmod{13} \). (e) \( x^2 \equiv 1 \pmod{8} \). (f) \( x^3 - x^2 + 2x - 2 \equiv 0 \pmod{11} \). (g) \( x \equiv 1 \pmod{5} \) and also \( x \equiv 2 \pmod{7} \). (Find all solutions modulo 35, that is, find the solutions satisfying \( 0 \le x \le 34 \).)
Transcribed Image Text:### 1.17. Find all values of \( x \) between 0 and \( m - 1 \) that are solutions of the following congruences. **Hint**: If you can’t figure out a clever way to find the solution(s), you can just substitute each value \( x = 1, x = 2, \ldots, x = m - 1 \) and see which ones work. (a) \( x + 17 \equiv 23 \pmod{37} \). (b) \( x + 42 \equiv 19 \pmod{51} \). (c) \( x^2 \equiv 3 \pmod{11} \). (d) \( x^2 \equiv 2 \pmod{13} \). (e) \( x^2 \equiv 1 \pmod{8} \). (f) \( x^3 - x^2 + 2x - 2 \equiv 0 \pmod{11} \). (g) \( x \equiv 1 \pmod{5} \) and also \( x \equiv 2 \pmod{7} \). (Find all solutions modulo 35, that is, find the solutions satisfying \( 0 \le x \le 34 \).)
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