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Difficulty: MediumGreatest Common Divisor (GCD) and Least Common Multiple (LCM)

A rectangular tabletop measuring 126 centimeters126\text{ centimeters} by 180 centimeters180\text{ centimeters} is to be completely covered by non-overlapping, identical square tiles of the maximum possible side length, such that no tiles need to be cut. How many such square tiles are required to cover the entire tabletop?

Answer: 70 tiles

Answer

70 tiles are required to cover the tabletop.
To cover a 126 cm×180 cm126\text{ cm} \times 180\text{ cm} surface with identical square tiles of maximum side length without cutting, the side length of each square tile must be the greatest common divisor of 126126 and 180180, which is 18 cm18\text{ cm}. Dividing the surface dimensions by 18 cm18\text{ cm} gives 77 tiles along one side and 1010 tiles along the other, yielding a total of 7070 tiles.

Step-by-Step Solution

1
Determine the prime factorizations of the tabletop dimensions 126 and 180.
126=21×32×71126 = 2^1 \times 3^2 \times 7^1 and 180=22×32×51180 = 2^2 \times 3^2 \times 5^1
Prime factorization allows systematic extraction of the greatest common divisor.
2
Calculate the Greatest Common Divisor (GCD) of 126 and 180.
GCD(126,180)=2min⁡(1,2)×3min⁡(2,2)=21×32=18\text{GCD}(126, 180) = 2^{\min(1,2)} \times 3^{\min(2,2)} = 2^1 \times 3^2 = 18
The maximum side length of a square tile that fits evenly without cutting is the GCD of the two dimensions.
3
Find the total number of tiles required.
\text{Total tiles} = \left(\frac{126}{18}\right) \times \left(\frac{180}{18}\right) = 7 \times 10 = 70
The total area divided by the area of one tile gives the number of tiles needed.

Key Concept

Greatest Common Divisor (GCD) application in geometric spatial partitioning
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Topics

Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

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