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Find an inverse for the linear congruence 9x = 15 (mod 23) by using the Euclidean algorithm to compute the gcd(9, 23) and performing backward substitution. Then find a solution for the linear congruence.

Find an inverse for the linear congruence 9x = 15 (mod 23) by using the Euclidean algorithm to compute the gcd(9, 23) and performing backward substitution. Then find a solution for the linear congruence.

Advanced Engineering Mathematics
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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### Title: Solving Linear Congruence Using the Euclidean Algorithm In this example, we aim to find an inverse for the linear congruence \(9x \equiv 15 \pmod{23}\) by using the Euclidean algorithm. We start by computing the gcd(9, 23) and then perform backward substitution. Finally, a solution will be determined for the linear congruence. 1. **Compute the gcd(9, 23) using the Euclidean algorithm:** - \(23 = 2 \cdot 9 + 5 \) - \(9 = 1 \cdot 5 + 4 \) - \(5 = 1 \cdot 4 + 1 \) - \(4 = 4 \cdot 1 + 0 \) Therefore, \(\text{gcd}(9, 23) = 1\). 2. **Backward substitution to express gcd as a linear combination:** - \(1 = 5 - 1 \cdot 4\) - \(4 = 9 - 1 \cdot 5\) Substituting \(4\), - \(1 = 5 - 1 \cdot (9 - 1 \cdot 5)\) - \(1 = 2 \cdot 5 - 1 \cdot 9\) Substituting \(5\), - \(5 = 23 - 2 \cdot 9\) Substituting \(5\), - \(1 = 2 \cdot (23 - 2 \cdot 9) - 1 \cdot 9\) - \(1 = 2 \cdot 23 - 5 \cdot 9\) 3. **Modular Inverse:** Find the inverse of \(9\) modulo \(23\), which can be seen from the linear combination as: - \(-5 \equiv 18 \pmod{23}\) 4. **Solution to the original congruence:** - Multiplying both sides by the inverse \(18\): \(x \equiv 15 \cdot 18 \pmod{23}\) - Calculate \(15 \cdot 18 \pmod{23}\) - \(15 \cdot
Transcribed Image Text:### Title: Solving Linear Congruence Using the Euclidean Algorithm In this example, we aim to find an inverse for the linear congruence \(9x \equiv 15 \pmod{23}\) by using the Euclidean algorithm. We start by computing the gcd(9, 23) and then perform backward substitution. Finally, a solution will be determined for the linear congruence. 1. **Compute the gcd(9, 23) using the Euclidean algorithm:** - \(23 = 2 \cdot 9 + 5 \) - \(9 = 1 \cdot 5 + 4 \) - \(5 = 1 \cdot 4 + 1 \) - \(4 = 4 \cdot 1 + 0 \) Therefore, \(\text{gcd}(9, 23) = 1\). 2. **Backward substitution to express gcd as a linear combination:** - \(1 = 5 - 1 \cdot 4\) - \(4 = 9 - 1 \cdot 5\) Substituting \(4\), - \(1 = 5 - 1 \cdot (9 - 1 \cdot 5)\) - \(1 = 2 \cdot 5 - 1 \cdot 9\) Substituting \(5\), - \(5 = 23 - 2 \cdot 9\) Substituting \(5\), - \(1 = 2 \cdot (23 - 2 \cdot 9) - 1 \cdot 9\) - \(1 = 2 \cdot 23 - 5 \cdot 9\) 3. **Modular Inverse:** Find the inverse of \(9\) modulo \(23\), which can be seen from the linear combination as: - \(-5 \equiv 18 \pmod{23}\) 4. **Solution to the original congruence:** - Multiplying both sides by the inverse \(18\): \(x \equiv 15 \cdot 18 \pmod{23}\) - Calculate \(15 \cdot 18 \pmod{23}\) - \(15 \cdot
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