These numbers are called "Highly Composite Numbers" [0]. Basically, it is a series of numbers where each number has more factors than the number before it (and is the first number with that number of factors). As you hinted, they are especially useful if your number system does not have fractions or decimal places and you still want to divide things.

You may recognize the beginning of the series: 1 2 4 6 12 24 36 48 60 120 180 240 360 720

Numberphile [1] calls them "Anti-Primes".

[0]: https://en.wikipedia.org/wiki/Highly_composite_number [1]: https://www.youtube.com/watch?v=2JM2oImb9Qg