Solve the equation x2≡180(mod209)�2≡180(mod209).

To solve the quadratic congruence \( x^2 \equiv 180 \pmod{209} \), we follow these steps:

### Step 1: Factor the modulus
We start by factoring the modulus 209:
\[
209 = 11 \times 19
\]
So, we'll solve the congruence modulo 11 and 19 separately, and then combine the results using the Chinese Remainder Theorem (CRT).

### Step 2: Solve the congruences modulo 11 and 19

1. **Solve \( x^2 \equiv 180 \pmod{11} \):**
   - First, reduce 180 modulo 11:
   \[
   180 \equiv 4 \pmod{11}
   \]
   - We need to find \( x \) such that \( x^2 \equiv 4 \pmod{11} \).
   - The possible values of \( x \) modulo 11 are \( x \equiv 2 \) and \( x \equiv 9 \) because:
   \[
   2^2 = 4 \quad \text{and} \quad 9^2 = 81 \equiv 4 \pmod{11}
   \]

2. **Solve \( x^2 \equiv 180 \pmod{19} \):**
   - Reduce 180 modulo 19:
   \[
   180 \equiv 9 \pmod{19}
   \]
   - We need to find \( x \) such that \( x^2 \equiv 9 \pmod{19} \).
   - The possible values of \( x \) modulo 19 are \( x \equiv 3 \) and \( x \equiv 16 \) because:
   \[
   3^2 = 9 \quad \text{and} \quad 16^2 = 256 \equiv 9 \pmod{19}
   \]

### Step 3: Combine solutions using the Chinese Remainder Theorem (CRT)
Now, we have four pairs of congruences to combine:

1. \( x \equiv 2 \pmod{11} \) and \( x \equiv 3 \pmod{19} \)
2. \( x \equiv 2 \pmod{11} \) and \( x \equiv 16 \pmod{19} \)
3. \( x \equiv 9 \pmod{11} \) and \( x \equiv 3 \pmod{19} \)
4. \( x \equiv 9 \pmod{11} \) and \( x \equiv 16 \pmod{19} \)

We'll solve each pair using CRT:

1. **For \( x \equiv 2 \pmod{11} \) and \( x \equiv 3 \pmod{19} \):**
   - The solution is \( x \equiv 111 \pmod{209} \).

2. **For \( x \equiv 2 \pmod{11} \) and \( x \equiv 16 \pmod{19} \):**
   - The solution is \( x \equiv 20 \pmod{209} \).

3. **For \( x \equiv 9 \pmod{11} \) and \( x \equiv 3 \pmod{19} \):**
   - The solution is \( x \equiv 191 \pmod{209} \).

4. **For \( x \equiv 9 \pmod{11} \) and \( x \equiv 16 \pmod{19} \):**
   - The solution is \( x \equiv 99 \pmod{209} \).

### Final Solutions:
The solutions to \( x^2 \equiv 180 \pmod{209} \) are:
\[
x \equiv 20, \, 99, \, 111, \, 191 \pmod{209}
\]

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Would you like further details on any step? Here are some additional questions:

1. How does the Chinese Remainder Theorem work?
2. What is the significance of quadratic residues in solving congruences?
3. How can you verify the solutions to the congruence?
4. What methods are available for solving quadratic congruences?
5. Why is factorization of the modulus important in this context?
6. How would the solution change if the modulus had a different factorization?
7. What are other applications of modular arithmetic?
8. Can you explain how the Legendre symbol helps in solving these types of problems?

**Tip:** Always check the consistency of your solutions when combining results from different moduli using the Chinese Remainder Theorem.

Learn how to solve the quadratic congruence x^2 ≡ 180 (mod 209) with a detailed step-by-step solution. This problem involves modular arithmetic, quadratic residues, and application of the Chinese Remainder Theorem. Suitable for advanced undergraduate students and beyond.

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