To solve this problem, we need to determine the size of the square tiles that can be used to pave the entire rectangular courtyard without any gaps or overlaps, and then calculate the number of such tiles required.
1. Convert the dimensions to a consistent unit:
- The length of the courtyard is [tex]\(18 \, \text{m} \, 72 \, \text{cm}\)[/tex].
[tex]\[ 18 \, \text{meters} = 18 \times 100 \, \text{cm} = 1800 \, \text{cm} \][/tex]
- Adding the extra 72 cm:
[tex]\[ 1800 \, \text{cm} + 72 \, \text{cm} = 1872 \, \text{cm} \][/tex]
- The width of the courtyard is [tex]\(13 \, \text{m} \, 20 \, \text{cm}\)[/tex].
[tex]\[ 13 \, \text{meters} = 13 \times 100 \, \text{cm} = 1300 \, \text{cm} \][/tex]
- Adding the extra 20 cm:
[tex]\[ 1300 \, \text{cm} + 20 \, \text{cm} = 1320 \, \text{cm} \][/tex]
2. Determine the largest possible size of the square tile:
- The size of the largest possible square tile that can perfectly fit both dimensions can be found by calculating the Greatest Common Divisor (GCD) of the two lengths. The GCD of 1872 cm and 1320 cm is 24 cm.
3. Calculate the number of square tiles needed:
- The area of the courtyard is:
[tex]\[ 1872 \, \text{cm} \times 1320 \, \text{cm} = 2471040 \, \text{cm}^2 \][/tex]
- The area of each square tile (with side length of 24 cm) is:
[tex]\[ 24 \, \text{cm} \times 24 \, \text{cm} = 576 \, \text{cm}^2 \][/tex]
- The number of tiles needed to pave the courtyard is:
[tex]\[ \frac{2471040 \, \text{cm}^2}{576 \, \text{cm}^2} = 4290 \][/tex]
Therefore, the least possible number of square tiles required to pave the courtyard is [tex]\(4290\)[/tex].
