To solve this problem, let's break it down step by step:

Part (a): Finding the largest size of the cubic boxes

The cubic boxes need to perfectly fit into the dimensions of the room. This means the side of the cube must be a divisor of the length, width, and height of the room. To find the largest possible size for the cubic boxes, we need to find the greatest common divisor (GCD) of the room's dimensions.

The room's dimensions are:

• Length = 560 cm
• Width = 350 cm
• Height = 420 cm

We need to calculate the GCD of 560, 350, and 420.

Step 1: Finding the GCD of 560 and 350

• 560 = 2³ × 5 × 7
• 350 = 2 × 5² × 7

The common factors are 2, 5, and 7. So, GCD(560, 350) = 2 × 5 × 7 = 70.

Step 2: Finding the GCD of 70 and 420

• 420 = 2² × 3 × 5 × 7

The common factors between 70 and 420 are 2, 5, and 7. So, GCD(70, 420) = 2 × 5 × 7 = 70.

Thus, the largest size of the cubic boxes is 70 cm.

Part (b): Calculating the number of boxes needed

The volume of the room and the volume of each cubic box will help us determine how many boxes will be used.

Step 1: Calculate the volume of the room

The volume of the room is: $\text{Volume of the room} = \text{Length} \times \text{Width} \times \text{Height}$ $\text{Volume of the room} = 560 \, \text{cm} \times 350 \, \text{cm} \times 420 \, \text{cm} = 82,320,000 \, \text{cm}^3$

Step 2: Calculate the volume of one cubic box

The side length of each cubic box is 70 cm, so the volume of one box is: $\text{Volume of one box} = 70 \, \text{cm} \times 70 \, \text{cm} \times 70 \, \text{cm} = 343,000 \, \text{cm}^3$

Step 3: Calculate the number of boxes

The number of boxes is the volume of the room divided by the volume of one cubic box: $\text{Number of boxes} = \frac{\text{Volume of the room}}{\text{Volume of one box}} = \frac{82,320,000}{343,000} = 240 \, \text{boxes}$

Final Answers:

• (a) The largest size of the cubic boxes is 70 cm.
• (b) The number of boxes needed to fill the room is 240.

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Do you want further details or have any questions about the solution?

Here are 5 related questions for further practice:

1. What would the size of the boxes be if the room dimensions were 600 cm, 400 cm, and 500 cm?
2. How does finding the GCD of numbers relate to solving problems like this?
3. If the cubic boxes were 50 cm in size, how many would be required to fill the same room?
4. How would the problem change if the room had different dimensions for each section (i.e., non-rectangular)?
5. What would the volume of the room be in cubic meters instead of cubic centimeters?

Tip: Finding the GCD is a useful technique for solving problems that involve fitting objects with specific dimensions into a given space.