A.  Choose two three-digit integers, a and b, with six distinct digits (e.g., a = 867 and b = 153; do not use these examples).

1.  Find the greatest common divisor, d, of the two numbers from part A, using the Euclidean algorithm. Show your work.

2.  Find all solutions for the congruence ax ≡ d (mod b), where a and b are the integers from part A and d is the greatest common divisor from part A1. Show your work.

Let's break this down step by step. 

### Step A: Choose two three-digit integers \( a \) and \( b \) with six distinct digits.
Let's choose \( a = 324 \) and \( b = 581 \). Both numbers have distinct digits, and no digits are repeated across the two numbers.

### Part 1: Find the Greatest Common Divisor (GCD) using the Euclidean Algorithm

The Euclidean algorithm for finding the GCD of two numbers involves repeatedly applying the division algorithm until the remainder is 0. 

1. Start with \( a = 324 \) and \( b = 581 \).
2. Apply the division algorithm to \( a \) and \( b \):
   \[
   581 = 324 \times 1 + 257
   \]
   Here, 257 is the remainder.
3. Replace \( a \) with \( b \) and \( b \) with the remainder, then repeat:
   \[
   324 = 257 \times 1 + 67
   \]
   Now, 67 is the remainder.
4. Continue the process:
   \[
   257 = 67 \times 3 + 56
   \]
   Now, 56 is the remainder.
   \[
   67 = 56 \times 1 + 11
   \]
   Now, 11 is the remainder.
   \[
   56 = 11 \times 5 + 1
   \]
   Now, 1 is the remainder.
   \[
   11 = 1 \times 11 + 0
   \]
   Here, the remainder is 0.

The GCD is the last non-zero remainder, which is \( \mathbf{d = 1} \).

### Part 2: Solve the Congruence \( 324x \equiv 1 \pmod{581} \)

We need to solve the congruence equation \( 324x \equiv 1 \pmod{581} \). Since the GCD of \( a \) and \( b \) is 1, a solution exists, and we can find it using the Extended Euclidean Algorithm.

Let's express 1 as a linear combination of 324 and 581:

From the Euclidean algorithm steps above:
\[
1 = 56 - 5 \times 11
\]
\[
1 = 56 - 5 \times (67 - 56)
\]
\[
1 = 6 \times 56 - 5 \times 67
\]
\[
1 = 6 \times (257 - 3 \times 67) - 5 \times 67
\]
\[
1 = 6 \times 257 - 23 \times 67
\]
\[
1 = 6 \times 257 - 23 \times (324 - 257)
\]
\[
1 = 29 \times 257 - 23 \times 324
\]
\[
1 = 29 \times (581 - 324) - 23 \times 324
\]
\[
1 = 29 \times 581 - 52 \times 324
\]

So, \( 324 \times (-52) \equiv 1 \pmod{581} \).

Thus, the solution is \( x \equiv -52 \pmod{581} \).

Since \( x \) needs to be positive, we add 581 to -52:

\[
x \equiv -52 + 581 \equiv 529 \pmod{581}
\]

So, the solution is \( \mathbf{x \equiv 529 \pmod{581}} \).

### Summary

1. The GCD \( d \) of \( a = 324 \) and \( b = 581 \) is \( \mathbf{1} \).
2. The solution to the congruence \( 324x \equiv 1 \pmod{581} \) is \( \mathbf{x \equiv 529 \pmod{581}} \).

Do you want further details or have any questions?

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**Related Questions:**
1. How does the Euclidean algorithm work?
2. What is the importance of the greatest common divisor?
3. How can we use the extended Euclidean algorithm to find modular inverses?
4. What is the modular arithmetic used for in cryptography?
5. What if the GCD was greater than 1, how would it affect the congruence equation?
6. How do you solve linear Diophantine equations?
7. What are some practical applications of finding GCD?
8. Can the Euclidean algorithm be used for more than two numbers?

**Tip:** The Euclidean algorithm is not only a fundamental tool in number theory but also a key concept in computer science for cryptographic algorithms.

Learn how to solve the congruence equation 324x ≡ 1 (mod 581) using the Euclidean algorithm and extended techniques. Understand the concept of modular arithmetic and its applications in number theory. Suitable for advanced high school students interested in number theory and algorithms.

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