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?

Question

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?

Answer 1

To find out the number of tiles required by the class, we need to determine the total number of tiles used to build the rectangles. Since each student had a different number of tiles and each student used all of their tiles to build a rectangle, we can find the total by summing up the number of tiles used by each student.

We know that there are 23 students in the class. Let's assume the number of tiles used by each student is n1, n2, n3, ..., n23.

To calculate the minimum number of rectangles used by the class, we need to find the prime factorization of the total number of tiles. This is because the minimum number of rectangles will occur when the total number of tiles is divided into the largest possible equal rectangular shapes.

Now, let's calculate the prime factorization of the total number of tiles used in the class.

First, calculate the sum of the number of tiles used by each student: n1 + n2 + n3 + ... + n23.

This sum will give us the total number of tiles required by the class.

Next, determine the prime factorization of this total number of tiles. This involves finding the prime numbers that divide exactly into the total number of tiles, and then expressing the total number of tiles as a product of these prime factors.

Once we have the prime factorization of the total number of tiles, we can determine the minimum number of rectangles by looking at the exponents of the prime factors.

For example, if the total number of tiles is 36, the prime factorization would be 2^2 * 3^2, which means that the minimum number of rectangles would be 2 * 2 * 3 * 3 = 4 * 9 = 36.

By following this process, you can determine the total number of tiles required by the class and calculate the minimum number of rectangles used to achieve this.