Use the Chinese Remainder Theorem to solve the system of conguences: x = 3 mod6 2x = 14 mod 20 I= 12 mod 45
Use the Chinese Remainder Theorem to solve the system of conguences: x = 3 mod6 2x = 14 mod 20 I= 12 mod 45
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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Use Chinese Remainder Theorem
![**Using the Chinese Remainder Theorem to Solve the System of Congruences**
We are given the following system of congruences:
1. \( x \equiv 3 \pmod{6} \)
2. \( 2x \equiv 14 \pmod{20} \)
3. \( x \equiv 12 \pmod{45} \)
To solve this system using the Chinese Remainder Theorem, follow these steps:
1. **Simplify Each Congruence:**
- The first congruence is already simplified:
\[
x \equiv 3 \pmod{6}
\]
- For the second congruence, divide the entire equation by 2 to simplify:
\[
x \equiv 7 \pmod{10}
\]
- The third congruence is already as simplified as possible:
\[
x \equiv 12 \pmod{45}
\]
2. **Ensure Coprimality of Moduli:**
The moduli should be pairwise coprime for the Chinese Remainder Theorem to apply directly.
- Check the pairwise coprimality:
- \(6\) and \(10\): The greatest common divisor (gcd) is \(2\).
- \(6\) and \(45\): The gcd is \(3\).
- \(10\) and \(45\): The gcd is \(5\).
In this case, the original moduli are not pairwise coprime, so the Chinese Remainder Theorem in its conventional form does not directly apply.
However, further algebraic methods or numerical checks can afford solutions in specific cases, or you could transform the congruences with GCD transformations and solve iteratively.
3. **Re-solve Where Problems Arise:**
You may need alternative modular arithmetic strategies or solving algorithms where traditional application cannot resolve such non-coprime modulus conditions.
This structured approach aligns well with educational platforms, offering clarity on how to proceed through such a problem iteratively or through strategic transformations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80fac62b-1f91-4b94-88b2-6d2c841a4645%2F850fda33-275a-4a06-b8a0-0ff751e5f464%2Fqie515a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using the Chinese Remainder Theorem to Solve the System of Congruences**
We are given the following system of congruences:
1. \( x \equiv 3 \pmod{6} \)
2. \( 2x \equiv 14 \pmod{20} \)
3. \( x \equiv 12 \pmod{45} \)
To solve this system using the Chinese Remainder Theorem, follow these steps:
1. **Simplify Each Congruence:**
- The first congruence is already simplified:
\[
x \equiv 3 \pmod{6}
\]
- For the second congruence, divide the entire equation by 2 to simplify:
\[
x \equiv 7 \pmod{10}
\]
- The third congruence is already as simplified as possible:
\[
x \equiv 12 \pmod{45}
\]
2. **Ensure Coprimality of Moduli:**
The moduli should be pairwise coprime for the Chinese Remainder Theorem to apply directly.
- Check the pairwise coprimality:
- \(6\) and \(10\): The greatest common divisor (gcd) is \(2\).
- \(6\) and \(45\): The gcd is \(3\).
- \(10\) and \(45\): The gcd is \(5\).
In this case, the original moduli are not pairwise coprime, so the Chinese Remainder Theorem in its conventional form does not directly apply.
However, further algebraic methods or numerical checks can afford solutions in specific cases, or you could transform the congruences with GCD transformations and solve iteratively.
3. **Re-solve Where Problems Arise:**
You may need alternative modular arithmetic strategies or solving algorithms where traditional application cannot resolve such non-coprime modulus conditions.
This structured approach aligns well with educational platforms, offering clarity on how to proceed through such a problem iteratively or through strategic transformations.
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