Find the greatest common divisor of 108 and 368

To find the greatest common divisor (GCD) of 108 and 368, we can use the Euclidean algorithm. Here's how it works:

1. **Divide 368 by 108** and find the remainder.
   \[
   368 \div 108 = 3 \text{ (quotient)} \quad \text{and} \quad 368 - 108 \times 3 = 44 \text{ (remainder)}
   \]

2. **Now, divide 108 by 44** and find the remainder.
   \[
   108 \div 44 = 2 \text{ (quotient)} \quad \text{and} \quad 108 - 44 \times 2 = 20 \text{ (remainder)}
   \]

3. **Next, divide 44 by 20** and find the remainder.
   \[
   44 \div 20 = 2 \text{ (quotient)} \quad \text{and} \quad 44 - 20 \times 2 = 4 \text{ (remainder)}
   \]

4. **Finally, divide 20 by 4** and find the remainder.
   \[
   20 \div 4 = 5 \text{ (quotient)} \quad \text{and} \quad 20 - 4 \times 5 = 0 \text{ (remainder)}
   \]

When the remainder becomes 0, the divisor at that step is the GCD. Here, the last non-zero remainder is 4.

So, the greatest common divisor of 108 and 368 is **4**.

Would you like more details or have any questions?

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Here are 8 related questions you might find interesting:

1. How do you find the least common multiple (LCM) of 108 and 368?
2. What are the prime factors of 108?
3. What are the prime factors of 368?
4. Can the Euclidean algorithm be used to find the GCD of three numbers?
5. What is the relationship between GCD and LCM of two numbers?
6. How would you express 368 as a product of its prime factors?
7. How can you use the GCD to simplify fractions?
8. How does the Euclidean algorithm work for very large numbers?

**Tip:** When dealing with large numbers, the Euclidean algorithm is more efficient than listing all divisors!

Learn how to find the greatest common divisor (GCD) of 108 and 368 using the Euclidean algorithm with a step-by-step explanation. Understand the basics of number theory and practical application of the Euclidean algorithm. Suitable for students in Grades 6-8.

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