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
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a453fbe-99a3-41fe-b2f6-1ca0d78d497c%2Fd4a9a67e-55b3-4191-9097-f89213e36186%2Fht42w5_processed.png&w=3840&q=75)
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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