A class of 23 students was using square tiles to build rectangular shapes. Each student had more than 1 tile and each had a different number of tiles. Each student was able to build only one shape of rectangle. All tiles had to be used to build a rectangle and the rectangle could not have holes. For example, a 2 by 6 uses 12 tiles and is consisted the same as a 6 by 2 rectangle but is different from a 3 by 4 rectangle. The class did the activity using the least number of tiles. How many tiles did the class use? Explore the various rectangles that could be made.
To solve this problem, we need to find a rectangle that can be built using the least number of tiles. Let's start exploring the possible rectangles that the class could have made:
1. We know that each student had more than 1 tile, so the minimum number of tiles per rectangle is 2.
- This means the smallest possible rectangle would be a 1 by 2, which uses 2 tiles.
2. For rectangles with more tiles, we need to consider the factors of the total number of tiles available.
- The class has a total of 23 students, and each student has a different number of tiles. This means the possible total number of tiles could be any unique sum of integers from 2 to 22.
3. We can start by considering rectangles that have dimensions with factors closest to each other.
- For example, for 2 tiles, we have a 1 by 2 rectangle.
- For 3 tiles, we have a 1 by 3 rectangle.
- For 4 tiles, we have a 1 by 4 rectangle, or a 2 by 2 rectangle.
4. As we increase the total number of tiles, we can continue to explore different combinations by finding pairs of factors that are close to each other.
- For example, for 5 tiles, we have a 1 by 5 rectangle or a 2 by 3 rectangle.
- For 6 tiles, we have a 2 by 3 rectangle or a 1 by 6 rectangle.
- For 7 tiles, we have a 1 by 7 rectangle.
5. We continue this process until we find a rectangle where the product of two factors equals the total number of tiles.
- For example, for 9 tiles, we have a 3 by 3 rectangle.
- For 10 tiles, we have a 2 by 5 rectangle.
6. We repeat steps 4 and 5 for all possible total numbers of tiles until we find a rectangle that can be built using the least number of tiles.
7. By exploring all possible rectangles, we find that the class would have needed a 4 by 6 rectangle, which uses 24 tiles. This rectangle is the only combination where no other rectangle can be formed using fewer tiles.
Therefore, the class would have used 24 tiles in total to build the rectangles.