Finding the Least Number of Square Tiles for Room Paving

The problem asks for the least number of square tiles required to cover the floor of a room that measures 15 meters by 12 meters. To use the least number of square tiles, each tile must be as large as possible. Since the tiles must be square and cover the entire floor without cutting, the side length of the square tile must be a common divisor of both the length (15 meters) and the width (12 meters) of the room. To get the least number of tiles, we need the largest possible square tile side length. This means the side length must be the Greatest Common Divisor (GCD) of the room's dimensions.

Calculating the Greatest Common Divisor (GCD)

The dimensions of the room are 15 meters and 12 meters. We need to find the GCD of 15 and 12. We can do this using prime factorization:

• Prime factorization of 15: $15 = 3 \times 5$
• Prime factorization of 12: $12 = 2 \times 2 \times 3 = 2^2 \times 3$

The common prime factors are multiplied to find the GCD. In this case, the only common prime factor is 3.

So, the GCD of 15 and 12 is 3.

This means the side length of the largest possible square tile that can perfectly tile the room is 3 meters.

Calculating the Number of Tiles

Now that we know the side length of the square tile is 3 meters, we can find the number of tiles needed. We can do this in two ways:

Method 1: Using Area

Calculate the area of the room and the area of one tile, then divide the room area by the tile area.

• Area of the room = Length $\times$ Width = $15 \text{ m} \times 12 \text{ m} = 180 \text{ m}^2$
• Area of one square tile = Side $\times$ Side = $3 \text{ m} \times 3 \text{ m} = 9 \text{ m}^2$
• Number of tiles = $\frac{\text{Area of room}}{\text{Area of one tile}} = \frac{180 \text{ m}^2}{9 \text{ m}^2}$

Number of tiles = $20$

Method 2: Counting Tiles Along Each Dimension

Calculate how many tiles fit along the length and how many fit along the width, then multiply these numbers.

• Number of tiles along the length = $\frac{\text{Room Length}}{\text{Tile Side Length}} = \frac{15 \text{ m}}{3 \text{ m}} = 5$ tiles
• Number of tiles along the width = $\frac{\text{Room Width}}{\text{Tile Side Length}} = \frac{12 \text{ m}}{3 \text{ m}} = 4$ tiles
• Total number of tiles = (Tiles along length) $\times$ (Tiles along width) = $5 \times 4$

Total number of tiles = $20$

Both methods show that 20 square tiles with a side length of 3 meters are needed to pave the room's floor using the least possible number of tiles.

Summary of Calculation

Therefore, the least number of square tiles required is 20.

Revision Table: Key Concepts for Tiling Problems

Additional Information: Understanding GCD in Real-World Problems

The Greatest Common Divisor (GCD) is a fundamental concept in number theory with many practical applications, such as this tiling problem. When you need to divide two quantities (like the dimensions of a room) into the largest possible equal parts (like the sides of square tiles) that fit exactly, the GCD is the key. In this case, finding GCD(15, 12) gives us 3, which is the largest possible side length for a square tile that will perfectly fit into a 15m x 12m room without any gaps or need for cutting tiles. Using a tile of any other size that is a common divisor (like 1m) would require more tiles (15x12 = 180 tiles of 1m side). Using a tile size that is not a common divisor would require cutting tiles.

Therefore, for problems asking for the minimum number of square units to cover a rectangular area without waste, always think of the GCD of the dimensions.