Find the greatest common divisor of 2947 and 3997. Indicate the property, definition, or theorem

To find the **Greatest Common Divisor (GCD)** of 2947 and 3997, we can use the **Euclidean algorithm**. This algorithm repeatedly applies the division algorithm to find the GCD. The key property used here is:

### **Theorem (Euclidean Algorithm):**  
For two integers \( a \) and \( b \) where \( a > b \), the greatest common divisor of \( a \) and \( b \) is the same as the greatest common divisor of \( b \) and \( a \mod b \). That is:  
\[
\gcd(a, b) = \gcd(b, a \mod b)
\]
We repeat this process until one of the numbers becomes 0, at which point the GCD is the other number.

Now let's apply the Euclidean algorithm to 2947 and 3997.

1. \( 3997 \div 2947 \) gives quotient 1, remainder \( 3997 - 1 \times 2947 = 1050 \).
   \[
   \gcd(3997, 2947) = \gcd(2947, 1050)
   \]

2. \( 2947 \div 1050 \) gives quotient 2, remainder \( 2947 - 2 \times 1050 = 847 \).
   \[
   \gcd(2947, 1050) = \gcd(1050, 847)
   \]

3. \( 1050 \div 847 \) gives quotient 1, remainder \( 1050 - 1 \times 847 = 203 \).
   \[
   \gcd(1050, 847) = \gcd(847, 203)
   \]

4. \( 847 \div 203 \) gives quotient 4, remainder \( 847 - 4 \times 203 = 35 \).
   \[
   \gcd(847, 203) = \gcd(203, 35)
   \]

5. \( 203 \div 35 \) gives quotient 5, remainder \( 203 - 5 \times 35 = 28 \).
   \[
   \gcd(203, 35) = \gcd(35, 28)
   \]

6. \( 35 \div 28 \) gives quotient 1, remainder \( 35 - 1 \times 28 = 7 \).
   \[
   \gcd(35, 28) = \gcd(28, 7)
   \]

7. \( 28 \div 7 \) gives quotient 4, remainder \( 28 - 4 \times 7 = 0 \).
   \[
   \gcd(28, 7) = 7
   \]

Thus, the **GCD of 2947 and 3997 is 7**.

Would you like to see any additional steps or explanations?

Here are some related questions:
1. Can the Euclidean algorithm be used for more than two numbers?
2. How do you find the least common multiple (LCM) using the GCD?
3. What is the relationship between the GCD and prime factorization?
4. How does the extended Euclidean algorithm work?
5. What is the GCD of two prime numbers?

**Tip:** The Euclidean algorithm is a fast and efficient method for computing the GCD, even for large numbers.

Find the greatest common divisor of 2947 and 3997. Indicate the property, definition, or theorem.

Learn how to find the greatest common divisor (GCD) of 2947 and 3997 using the Euclidean Algorithm. This step-by-step guide covers essential number theory concepts and is suitable for high school students in Grades 9-12.

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