Numbers that have an odd number of factors are perfect squares. This might seem a bit surprising at first, but let’s break it down.
Every number has factors that typically come in pairs. For instance, if you take the number 12, its factors are 1, 2, 3, 4, 6, and 12. These can be paired as (1, 12), (2, 6), and (3, 4). Each pair multiplies to the original number. However, when it comes to perfect squares, like 36, one of the factors (in this case, 6) pairs with itself. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36, where 6 is the midpoint.
This self-pairing creates one unpaired factor, causing perfect squares to have an odd total of factors. Therefore, only perfect square numbers—such as 1, 4, 9, 16, and so on—end up with an odd number of factors.