To find the greatest common divisor (GCD) using the Euclidean algorithm, we repeatedly apply the rule that the GCD of two numbers also divides their difference. Here’s how to apply it to the given pairs:

1. GCD of 42 and 724

1. Divide 724 by 42: $724 \div 42 = 17 \quad \text{(remainder 10)}$ So, $724 = 42 \times 17 + 10$.

2. Now apply the algorithm to 42 and 10: $42 \div 10 = 4 \quad \text{(remainder 2)}$ So, $42 = 10 \times 4 + 2$.

3. Next, apply it to 10 and 2: $10 \div 2 = 5 \quad \text{(remainder 0)}$ So, $10 = 2 \times 5 + 0$.

When we reach a remainder of 0, the last non-zero remainder is the GCD. Thus, $\text{GCD}(42, 724) = 2$.

2. GCD of 1512 and 3228

1. Divide 3228 by 1512: $3228 \div 1512 = 2 \quad \text{(remainder 204)}$ So, $3228 = 1512 \times 2 + 204$.

2. Now apply the algorithm to 1512 and 204: $1512 \div 204 = 7 \quad \text{(remainder 144)}$ So, $1512 = 204 \times 7 + 144$.

3. Next, apply it to 204 and 144: $204 \div 144 = 1 \quad \text{(remainder 60)}$ So, $204 = 144 \times 1 + 60$.

4. Now to 144 and 60: $144 \div 60 = 2 \quad \text{(remainder 24)}$ So, $144 = 60 \times 2 + 24$.

5. Next, apply it to 60 and 24: $60 \div 24 = 2 \quad \text{(remainder 12)}$ So, $60 = 24 \times 2 + 12$.

6. Now to 24 and 12: $24 \div 12 = 2 \quad \text{(remainder 0)}$ So, $24 = 12 \times 2 + 0$.

Again, we reach a remainder of 0, so the GCD is the last non-zero remainder. Thus, $\text{GCD}(1512, 3228) = 12$.

Summary

• $\text{GCD}(42, 724) = 2$
• $\text{GCD}(1512, 3228) = 12$

If you need further details or explanations, feel free to ask! Here are some related questions you might consider:

1. How is the Euclidean algorithm derived?
2. Can the Euclidean algorithm be used for more than two numbers?
3. What is the significance of the GCD in number theory?
4. How does the GCD relate to fractions?
5. Are there any applications of the GCD in computer science?

Tip: Practicing the Euclidean algorithm with different pairs of integers can help reinforce your understanding of the process!