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Difficulty: MediumFactors, Multiples, and Prime Factorization

A rectangular courtyard measures 84 feet by 120 feet. A contractor wants to pave the courtyard using the fewest possible identical square stone tiles such that no tiles are cut and the entire courtyard is covered. What is the total number of square tiles the contractor will need?

  1. A
    12
  2. B
    17
  3. 70Answer
  4. D
    280
  5. E
    840

Answer

70
To find the fewest number of identical square tiles that can cover the courtyard without being cut, we must find the largest possible size for the square tiles. The side length of the square tiles must be a common factor of the dimensions of the courtyard, 84 and 120. To minimize the number of tiles, we maximize the tile size by finding the greatest common factor (GCF) of 84 and 120.

The prime factorization of 84 is 22×3×72^2 \times 3 \times 7.
The prime factorization of 120 is 23×3×52^3 \times 3 \times 5.

The GCF is 22×3=122^2 \times 3 = 12.

Thus, each square tile has a side length of 12 feet. Next, we determine how many tiles fit along each dimension of the courtyard:
Along the 84-foot side: 84÷12=784 \div 12 = 7 tiles.
Along the 120-foot side: 120÷12=10120 \div 12 = 10 tiles.

The total number of square tiles needed is the product of these two quantities: 7×10=707 \times 10 = 70 tiles.

Step-by-Step Solution

1
Find the greatest common factor (GCF) of the two dimensions, 84 and 120, to determine the maximum side length of the square tiles.
The GCF of 84 and 120 is 12, meaning each square tile will have a side length of 12 feet.
To cover the courtyard with the fewest possible square tiles without cutting, the tiles must be as large as possible, so their side length must be the greatest common divisor of the courtyard's dimensions.
2
Calculate the number of tiles needed along the length and the width of the courtyard.
7 tiles fit along the 84-foot side (84÷12=784 \div 12 = 7), and 10 tiles fit along the 120-foot side (120÷12=10120 \div 12 = 10).
This determines the grid dimensions of the tiles covering the rectangular floor.
3
Multiply the number of tiles along the length by the number of tiles along the width to find the total number of tiles.
7×10=707 \times 10 = 70 tiles.
The total number of tiles is the product of the grid dimensions.

Key Concept

Using the Greatest Common Factor (GCF) to solve real-world optimization and division problems.
Estimated Time:1m 30s
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Topics

Factors, Multiples, and Prime Factorization

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