give me some euclidean/extended euclidean questions

Here are some Euclidean and Extended Euclidean Algorithm questions that can help deepen your understanding:

1. **Find the GCD using the Euclidean Algorithm:**
   Compute the greatest common divisor (GCD) of 252 and 105 using the Euclidean Algorithm.

2. **Extended Euclidean Algorithm to find coefficients:**
   Use the Extended Euclidean Algorithm to express the GCD of 56 and 15 as a linear combination (i.e., find integers \( x \) and \( y \) such that \( 56x + 15y = \text{GCD}(56, 15) \)).

3. **Solve Diophantine Equations using Extended Euclidean:**
   Solve the equation \( 99x + 78y = 3 \) using the Extended Euclidean Algorithm. Find all integer solutions.

4. **Modular Inverse with Extended Euclidean:**
   Find the modular inverse of 17 modulo 43 using the Extended Euclidean Algorithm.

5. **General GCD Calculation:**
   Compute the GCD of 987 and 610 using the Euclidean Algorithm and verify that it follows the properties of Fibonacci numbers.

### Tip:
In the Extended Euclidean Algorithm, once you have the GCD, you can work backwards to find the coefficients for the linear combination (Bézout's identity).

Let me know if you want detailed solutions or further explanations!

Here are 5 more questions that you can explore:
1. Apply the Euclidean Algorithm to find the GCD of 123 and 321.
2. Solve \( 112x + 75y = 1 \) using the Extended Euclidean Algorithm.
3. How can the Extended Euclidean Algorithm be used to solve linear congruences?
4. Find the GCD of 120 and 35, and express it as a linear combination.
5. Why is the Extended Euclidean Algorithm useful for cryptography, particularly in RSA?



Explore a range of Euclidean and Extended Euclidean Algorithm problems, including GCD calculations, solving linear Diophantine equations, and finding modular inverses. Perfect for students in Grades 9-12 looking to deepen their understanding of number theory.

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