Question

The dimensions of a floor are 18 x 24. What is the smallest number of identical square tiles that pave the entire floor without the need to break any tile?

• A. 6
• B. 24
• C. 8
• D. 12

Answer 1

The Correct Option is D

To find the smallest number of identical square tiles that can pave the entire floor without breaking any tiles, we need to determine the largest size of such a square tile that can fit both dimensions of the floor without leaving any remainder. This is equivalent to finding the greatest common divisor (GCD) of the floor's dimensions.

Given the floor dimensions:

• Length = 18 units
• Width = 24 units 

Step 1: Calculate the GCD of the dimensions.

We can use the Euclidean algorithm to find the GCD:

• \(24 \div 18 = 1\) remainder \(6\)
• \(18 \div 6 = 3\) remainder \(0\)

Since the remainder is now 0, the GCD is the last non-zero remainder, which is \(6\).

Step 2: Calculate how many tiles are needed along each dimension using the GCD.

• Number of tiles along the length: \(\frac{18}{6} = 3\)
• Number of tiles along the width: \(\frac{24}{6} = 4\)

Total number of tiles needed:

• \(3 \times 4 = 12\)

Therefore, the smallest number of identical square tiles needed to pave the entire floor is 12.

The correct answer is option:

12