A class of 23 students was using square tiles to build rectangular shapes. Each student has more than one tile and each has a different number of tiles. Each student was able to build only one rectangular shape. All tiles had to be used to build the rectangle and the rectangle could have no holes. For example, a 2 by 6 rectangle uses 12 tiles and is considered 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 rectangles. How many tiles did the class require?
if only one shape can be built, then that means each student's number of tiles was prime. Any non-prime has at least two sets of factors.
so, what's the sum of the first 23 primes?
these nut
To find out how many tiles the class required, we need to determine the dimensions of the rectangular shapes built by each student. We know that each student used a different number of tiles, and all the tiles had to be used to build the rectangles without any holes.
Since there are 23 students in the class, we can assume that the smallest number of tiles used by a student is 1, and the largest number of tiles used is 23. Therefore, we need to find a pair of factors that multiply to give us a number between 1 and 23.
Starting from the smallest possible dimension, we can consider a rectangle with dimensions 1 by 23, which uses a total of 1 tile. However, we already know that each student used more than one tile, so this is not a valid solution.
Next, we can consider a rectangle with dimensions 2 by 12, which also uses a total of 24 tiles (2 x 12 = 24). Since the number of tiles used is greater than 23, this solution is not valid either.
Continuing this process, we can check other dimensions until we find a valid solution. We can see that a rectangle with dimensions 3 by 8 uses a total of 24 tiles (3 x 8 = 24). This is the first valid solution we have found.
However, we need to check if there is a smaller rectangle that can be built using fewer tiles. In this case, we can see that a rectangle with dimensions 4 by 6 also uses a total of 24 tiles (4 x 6 = 24). Since this is the same number of tiles as the previous solution, we have found another valid solution.
So far, we have found two valid solutions: a 3 by 8 rectangle and a 4 by 6 rectangle, both using 24 tiles. But we need to find the solution that requires the least number of rectangles.
To do this, we need to find the factors of 24 and check if any of them can be valid dimensions for a rectangle. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Checking each factor, we can see that no other dimensions result in a valid solution. Therefore, the class required 24 tiles in total.
In summary, the class of 23 students required a total of 24 tiles to build rectangular shapes, with the options of either a 3 by 8 rectangle or a 4 by 6 rectangle.