#math trivia for #April14: #105 sets a new record for number of odd divisors. What is it and when will it be broken again?

— Burt Kaliski Jr. (@modulomathy) April 14, 2012

There are eight odd divisors of 105: 1, 3, 5, 7, 15, 21, 35 and 105. The total of eight makes sense given that there are three prime factors, none repeated, so each one can be either included or not in a divisor: 2*2*2 = 8.

The record will be broken next when one of two things happens:

- A number has three unique prime factors and one is repeated, which means there are 3*2*2 = 12 divisors.
- A number has two unique prime factors and both are repeated, which means there are 3*3 = 9 divisors.

The smallest number of the first type is 3*3*5*7 = 345.

The smallest of the second type is 3*3*5*5 = 225. That will be the next one to break the record for odd divisors. (The nine odd divisors of 225 are: 1, 3, 5, 9, 15, 25, 45, 75, and 225.)