Solve the equation x2≡415(mod437)
The solutions are:

The equation \( x^2 \equiv 415 \pmod{437} \) is a quadratic congruence. To solve this, we'll determine the values of \( x \) such that \( x^2 \equiv 415 \pmod{437} \).

Here's how to approach it:

1. **Check if the congruence has a solution**: First, we need to check whether 415 is a quadratic residue modulo 437. This can be done using the Legendre symbol or other methods, but here we proceed directly by checking possible squares.

2. **Factorization**: Factor 437 to apply the Chinese Remainder Theorem if needed. 
   - \( 437 = 19 \times 23 \)

3. **Solve the congruences modulo 19 and 23**:
   - \( x^2 \equiv 415 \pmod{437} \) can be split into two congruences:
     - \( x^2 \equiv 415 \pmod{19} \)
     - \( x^2 \equiv 415 \pmod{23} \)

   But \( 415 \equiv 17 \pmod{19} \) and \( 415 \equiv 1 \pmod{23} \), so the problem reduces to solving:
   - \( x^2 \equiv 17 \pmod{19} \)
   - \( x^2 \equiv 1 \pmod{23} \)

4. **Find solutions to each**:
   - For \( x^2 \equiv 17 \pmod{19} \):
     - We check all possible values of \( x \) from 0 to 18. The solutions are \( x \equiv 7 \) and \( x \equiv 12 \).
   - For \( x^2 \equiv 1 \pmod{23} \):
     - The solutions are \( x \equiv 1 \) and \( x \equiv 22 \).

5. **Combine using the Chinese Remainder Theorem**: We combine the solutions for the two congruences:
   - Combine \( x \equiv 7 \pmod{19} \) with \( x \equiv 1 \pmod{23} \).
   - Combine \( x \equiv 7 \pmod{19} \) with \( x \equiv 22 \pmod{23} \).
   - Combine \( x \equiv 12 \pmod{19} \) with \( x \equiv 1 \pmod{23} \).
   - Combine \( x \equiv 12 \pmod{19} \) with \( x \equiv 22 \pmod{23} \).

These combinations will give us the solutions modulo 437.

### The final solutions are:
\[ x \equiv 270 \pmod{437} \]
\[ x \equiv 167 \pmod{437} \]
\[ x \equiv 375 \pmod{437} \]
\[ x \equiv 62 \pmod{437} \]

So the solutions to \( x^2 \equiv 415 \pmod{437} \) are \( x \equiv 62 \), \( 167 \), \( 270 \), and \( 375 \) modulo 437.

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Would you like any details on the steps? Here are some additional questions:

1. How is the Chinese Remainder Theorem applied in this context?
2. What is the significance of quadratic residues in modular arithmetic?
3. How would you verify if a number is a quadratic residue modulo another number?
4. Can you solve similar congruences for different moduli?
5. What are Legendre symbols, and how do they help in finding quadratic residues?
6. How do we factorize large numbers for applications like this?
7. Can you explain how solutions to modular arithmetic can be checked?
8. What is the relevance of modular arithmetic in cryptography?

**Tip:** In modular arithmetic, a number \( a \) is a quadratic residue modulo \( n \) if there exists an integer \( x \) such that \( x^2 \equiv a \pmod{n} \).

Learn how to solve the quadratic congruence x^2 ≡ 415 (mod 437) with detailed step-by-step solutions. This problem involves modular arithmetic concepts such as quadratic residues and the application of the Chinese Remainder Theorem. Suitable for advanced high school students and beyond.

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