Question

The painting will be made of 24cm by 32cm rectangles covered with colored squares. They want these squares to be as large as possible, measured in whole numbers of centimeters.

Then they plan to arrange copies of the 24cm by 32cm rectangle to form a large square painting that measures the least possible whole number of centimeters.
What size will the small colored squares be?

Answer 1

The size of the small colored squares that fit into a 24cm by 32cm rectangle will be 8 cm on each side, determined by the greatest common divisor of 24 and 32. To form a large square painting, the side length will be 96 cm, found by calculating the least common multiple of 24 and 32.

Step-by-step explanation:

The question asks about finding the size of the largest possible squares that can fit within a rectangle with dimensions of 24 cm by 32 cm, and then using copies of this rectangle to form the smallest possible whole number length square painting. The size of the squares is determined by the greatest common divisor (GCD) between the length and width of the rectangle. To find the GCD of 24 cm and 32 cm, we list the divisors of each:

• Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Divisors of 32: 1, 2, 4, 8, 16, 32

The largest common divisor is 8 cm, so the colored squares will be 8 cm on each side. Then, to make a large square painting using several 24cm by 32cm rectangles, we need to find the least number of these rectangles that can form a square. This can be done by finding the least common multiple (LCM) of 24 and 32, which is 96. Therefore, the side of the large square will be 96 cm, which is the smallest number that both 24 and 32 divide into without leaving a remainder.