A rectangular floor measuring 126\text{ cm} by 180\text{ cm} is to be completely covered w… — GRE General Test

A rectangular floor measuring $126\text{ cm}$ by $180\text{ cm}$ is to be completely covered with identical square tiles of the largest possible side length, without cutting any tiles or leaving gaps. What is the total number of square tiles required to cover the floor?

A
18
B
35
70Cevap
D
140
E
306

Cevap

The total number of square tiles required is 70.

To cover the rectangular floor with the largest possible identical square tiles, the side length of the square tile must be the greatest common divisor of

126

and

180

. Factoring both numbers gives

126 = 2 \times 3^2 \times 7

and

180 = 2^2 \times 3^2 \times 5

. The greatest common divisor is

2 \times 3^2 = 18\text{ cm}

. Dividing the dimensions by

18\text{ cm}

gives

7

tiles along the length and

10

tiles along the width. Multiplying

7 \times 10

yields a total of

70

tiles.

Adım Adım Çözüm

Find the prime factorization of both dimensions of the floor.

126 = 2^1 \times 3^2 \times 7^1

and

180 = 2^2 \times 3^2 \times 5^1

Prime factorization allows systematic determination of the greatest common divisor.

Calculate the Greatest Common Divisor (GCD) of 126 and 180 to determine the largest possible square tile side length.

\gcd(126, 180) = 2^1 \times 3^2 = 18\text{ cm}

The square tiles must fit evenly along both the length and width without cutting.

Determine the number of tiles along each dimension.

Along length:

126 \div 18 = 7

; along width:

180 \div 18 = 10

Dividing each total length by the tile side length gives the tile count per side.

Calculate the total number of tiles required.

7 \times 10 = 70\text{ tiles}

The floor forms a 7 by 10 grid of tiles.

Anahtar Kavram

Greatest Common Divisor (GCD) applications in geometric tiling

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