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The problem provided in the image asks to find the minimum possible length of a room given the following information:

- A person covers the floor using rectangular tiles along the length of the room.
- When tiles of length 48 cm are used, 10 cm of space is left at the end of a row.
- The same happens when tiles of length 80 cm are used (10 cm left over).

### Solution:

We are tasked with finding the minimum possible length \( L \) of the room.

Let the length of the room be \( L \). According to the problem, when using tiles of length 48 cm and 80 cm, there is always 10 cm left after laying down the tiles along the room's length. This means that the length of the room minus 10 cm is divisible by both 48 cm and 80 cm.

Mathematically, this can be represented as:
\[
L - 10 \, \text{cm} \, = \, \text{LCM}(48 \, \text{cm}, \, 80 \, \text{cm})
\]
where LCM is the least common multiple.

#### Step 1: Find the LCM of 48 and 80.

The prime factorization of 48 and 80 is:
\[
48 = 2^4 \times 3
\]
\[
80 = 2^4 \times 5
\]
The LCM takes the highest powers of all prime factors:
\[
\text{LCM}(48, 80) = 2^4 \times 3 \times 5 = 240
\]

#### Step 2: Calculate the minimum possible length of the room.

Using the LCM result:
\[
L - 10 = 240 \, \text{cm}
\]
\[
L = 240 + 10 = 250 \, \text{cm}
\]

Thus, the minimum possible length of the room is **250 cm**.

Let me know if you'd like any further details or clarification!

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Here are 5 related questions for further understanding:

1. What is the general method for calculating the least common multiple (LCM) of two numbers?
2. How can prime factorization help in determining the LCM of two numbers?
3. What would happen if the leftover space was different for the two types of tiles?
4. Could we solve this problem using any other mathematical technique like linear Diophantine equations?
5. How would the result change if the tiles were laid along the width instead of the length?

**Tip:** When working with problems involving divisibility or remainders, LCM often plays a key role in finding common patterns or solutions.

A person is covering the floor of a room using rectangular tiles by laying them along the length of the room. When he used tiles of length 48 cm, 10 cm of space was left at the end of a row. The same thing happened when he used tiles of length 80 cm. What is the minimum possible length of the room? Show your work.

Solve the problem of finding the minimum possible length of a room when using rectangular tiles of lengths 48 cm and 80 cm, leaving a 10 cm gap. Learn the process of using the least common multiple (LCM) to solve this problem, suitable for students in Grades 6-8.

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