Let's work through the problem in the image together.

Problem Breakdown:

1. Quilt Dimensions: The quilt is 60 inches by 75 inches. We need to figure out how big each square should be, given that we're using the largest possible square to fit into the quilt.

   To do this, we need to find the greatest common divisor (GCD) of 60 and 75. The GCD will tell us the size of the largest square that can fit evenly into both dimensions.

   Step 1: Find the GCD of 60 and 75

   • Prime factorization of 60: $60 = 2^2 \times 3 \times 5$
   • Prime factorization of 75: $75 = 3 \times 5^2$
   • The common factors are 3 and 5, and their product is 15.

   Answer: The largest square will have a side length of 15 inches.

2. How many squares will fit? Now that we know each square is 15 inches by 15 inches, we can calculate how many such squares fit into the quilt.

   Step 2: Calculate the number of squares

   • Number of squares along the 60-inch side: $\frac{60}{15} = 4$
   • Number of squares along the 75-inch side: $\frac{75}{15} = 5$
   • Total number of squares = $4 \times 5 = 20$

   Answer: There will be 20 squares in the quilt.

3. Why do we not find a Least Common Factor (LCF) and Greatest Common Multiple (GCM)? The Least Common Factor is always 1 for any set of integers, and the Greatest Common Multiple (GCM) refers to the smallest number that is a multiple of the given numbers (60 and 75). However, in this problem, we are looking to divide the quilt into squares, which is why we find the GCD. The GCD represents the largest dimension of the square that can divide both dimensions of the quilt evenly, and this is why finding the LCF and GCM would not be useful in this case.

Would you like more details or have any questions?

Here are 5 related questions to think about:

1. How would the answer change if the quilt was 48 inches by 72 inches?
2. What if we used rectangles instead of squares? How would we approach the problem?
3. Can you think of a situation where finding the LCM (Least Common Multiple) would be useful?
4. What are other applications of the GCD in real-life situations?
5. How would you explain the concept of GCD and LCM to someone new to math?

Tip: Whenever you're faced with a situation requiring cutting or dividing things into equal parts, think about using the GCD to find the largest possible division!