How Many Rectangles Can You Build With a Prime Number of Square Tiles?

By Staff WriterLast Updated December 05, 2023

Assuming orientation doesn’t matter, the number of rectangles that can be made from any particular prime number of square tiles is one. Prime numbers are only divisible by one and themselves, so there is only one possible way of laying out tiles in a rectangular formation.

A rectangle made from 13 tiles, for example, would have the dimensions 13 by 1. This way of making rectangles could lead to some very long, very skinny rectangles, but even an array many millions of tiles long would still have a width equal to one tile and be a valid rectangle. Composite numbers have at least three factors including one and themselves, so at least two distinct rectangular arrays must be possible for any composite number.