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2x = 1(mod 5) 31( mod 7) 5x 9(mod l)

2x = 1(mod 5) 31( mod 7) 5x 9(mod l)

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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Find all solutions using Chinese Reminder Theorem.
### System of Congruences Given the following system of congruences: 1. \(2x \equiv 1 \pmod{5}\) 2. \(4x \equiv 1 \pmod{7}\) 3. \(5x \equiv 9 \pmod{11}\) These equations represent a system of linear congruences. Each congruence is an equation that must hold true for integer values of \(x\). - The first congruence states that \(2x\) is congruent to 1 modulo 5, meaning that when \(2x\) is divided by 5, the remainder is 1. - The second congruence states that \(4x\) is congruent to 1 modulo 7. - The third congruence states that \(5x\) is congruent to 9 modulo 11. The goal is to find values of \(x\) that satisfy all three congruences simultaneously. This can often be solved using the Chinese Remainder Theorem, assuming the moduli (5, 7, 11) are coprime.
Transcribed Image Text:### System of Congruences Given the following system of congruences: 1. \(2x \equiv 1 \pmod{5}\) 2. \(4x \equiv 1 \pmod{7}\) 3. \(5x \equiv 9 \pmod{11}\) These equations represent a system of linear congruences. Each congruence is an equation that must hold true for integer values of \(x\). - The first congruence states that \(2x\) is congruent to 1 modulo 5, meaning that when \(2x\) is divided by 5, the remainder is 1. - The second congruence states that \(4x\) is congruent to 1 modulo 7. - The third congruence states that \(5x\) is congruent to 9 modulo 11. The goal is to find values of \(x\) that satisfy all three congruences simultaneously. This can often be solved using the Chinese Remainder Theorem, assuming the moduli (5, 7, 11) are coprime.
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