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The image contains Exercise 5.2 from a math textbook. The problems are focused on finding the greatest common divisor (GCD) and least common multiple (LCM), as well as solving for the largest numbers or measurements that satisfy specific division or remainder conditions.

### Let's solve the first question:

1. **Find the greatest number that will divide 45 and 105 exactly.**
   - This requires finding the GCD (Greatest Common Divisor) of 45 and 105.

To find the GCD, we can use the prime factorization method:
- Prime factorization of 45 = \( 3^2 \times 5 \)
- Prime factorization of 105 = \( 3 \times 5 \times 7 \)

The common factors are \( 3 \times 5 = 15 \), so the GCD is **15**.

Would you like me to solve other questions in the exercise or explain any details further?

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Here are 5 questions expanding on the concept of GCD and LCM:
1. What is the difference between GCD and LCM, and how are they applied in real-world problems?
2. Can the GCD of two numbers ever be greater than one of the numbers?
3. How can we use the Euclidean algorithm to find the GCD of two numbers more efficiently?
4. What is the relationship between the GCD and LCM of two numbers?
5. How does the remainder affect GCD when solving problems involving division with remainders?

### Tip:
When finding the GCD, if you're working with larger numbers, the Euclidean algorithm is much faster than prime factorization!

Find the greatest number that will divide 45 and 105 exactly.

This page provides a detailed explanation of how to find the greatest common divisor (GCD) of 45 and 105 using the prime factorization method. Learn how to break down each number and apply GCD concepts, suitable for students in Grades 6-8.

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