How many rectangles can you build with a prime number of square tiles

for n > 2 ... (n - 1) / 2

12 To determine the number of rectangles that can be built with a prime number of square tiles, we need to understand the properties of prime numbers and rectangles.

A prime number is a positive integer greater than 1 that has no divisors other than 1 and itself. For example, 2, 3, 5, and 7 are prime numbers.

A rectangle is a quadrilateral with four right angles. It has two pairs of opposite sides that are equal in length. The number of square tiles required to build a rectangle is equal to the product of its length and width.

To count the number of rectangles with a prime number of square tiles, we first need to determine the possible combinations of lengths and widths. We consider only pairs of integers that multiply to give a prime number.

For instance, let's consider the prime number 7. The integer pairs that multiply to give 7 are (1, 7) and (7, 1). In this case, we have two possible rectangles with 7 square tiles: one with a length of 1 and a width of 7, and another with a length of 7 and a width of 1.

Similarly, for the prime number 5, the integer pairs that multiply to give 5 are (1, 5) and (5, 1). So, there are two possible rectangles with 5 square tiles: one with a length of 1 and a width of 5, and another with a length of 5 and a width of 1.

In general, for any prime number p, there will be two possible rectangles: one with a length of 1 and a width of p, and another with a length of p and a width of 1.

Therefore, the number of rectangles that can be built with a prime number of square tiles is always 2.

Keep in mind that this explanation assumes square tiles and does not consider rotations or reflections of rectangles. For irregular-shaped tiles or more complex cases, the calculation of rectangles might involve additional considerations.