To find the greatest common divisor (gcd) of the numbers 970, 485, and 1330, we can use the Euclidean algorithm. The gcd is the largest number that divides all three numbers without leaving a remainder.

### Step 1: Find the gcd of 970 and 485
1. Divide 970 by 485:
   \[
   970 \div 485 = 2 \quad \text{(quotient)} \quad \text{remainder} = 970 - 485 \times 2 = 970 - 970 = 0
   \]
   So, \( \gcd(970, 485) = 485 \).

### Step 2: Find the gcd of 485 and 1330
2. Divide 1330 by 485:
   \[
   1330 \div 485 = 2 \quad \text{(quotient)} \quad \text{remainder} = 1330 - 485 \times 2 = 1330 - 970 = 360
   \]
3. Now, find the gcd of 485 and 360:
   \[
   485 \div 360 = 1 \quad \text{(quotient)} \quad \text{remainder} = 485 - 360 \times 1 = 485 - 360 = 125
   \]
4. Find the gcd of 360 and 125:
   \[
   360 \div 125 = 2 \quad \text{(quotient)} \quad \text{remainder} = 360 - 125 \times 2 = 360 - 250 = 110
   \]
5. Find the gcd of 125 and 110:
   \[
   125 \div 110 = 1 \quad \text{(quotient)} \quad \text{remainder} = 125 - 110 \times 1 = 125 - 110 = 15
   \]
6. Find the gcd of 110 and 15:
   \[
   110 \div 15 = 7 \quad \text{(quotient)} \quad \text{remainder} = 110 - 15 \times 7 = 110 - 105 = 5
   \]
7. Find the gcd of 15 and 5:
   \[
   15 \div 5 = 3 \quad \text{(quotient)} \quad \text{remainder} = 15 - 5 \times 3 = 15 - 15 = 0
   \]
   So, \( \gcd(15, 5) = 5 \).

### Final Result:
Thus, the gcd of 970, 485, and 1330 is **5**.

Would you like more details on the Euclidean algorithm, or do you have any further questions?

### Related questions:
1. How does the Euclidean algorithm work step by step?
2. How do you find the gcd of more than two numbers?
3. What is the connection between the gcd and the least common multiple (LCM)?
4. How can you use the gcd to simplify fractions?
5. Can the gcd of two numbers be greater than one without the numbers sharing a common factor?

**Tip**: The Euclidean algorithm can be applied to any pair of numbers and is very efficient for finding the gcd.

Learn how to find the greatest common divisor (GCD) of the numbers 970, 485, and 1330 using the Euclidean algorithm. This problem walks through each step in the process, including division and remainder calculations. Ideal for students in Grades 9-11 studying number theory and the Euclidean algorithm.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation