What number has exactly 100 divisors?

Answer 1

This is an aspect of number theory but let's start with a rule and a simple example.

If if it just a total number of factors that is required (rather than the values) then this can be obtained by stating the original number in terms of its prime factors.

For example , 100 = 22 x 52.

Add 1 to each exponent and obtain the product : (2 + 1)(2 + 1) = 3 x 3 = 9

Then 100 has 9 factors which are 100, 50, 25, 20, 10, 5, 4, 2, 1.

So for a number to have 100 factors then its prime factors must have exponents of 49 & 1, or 24 & 3, or 19 & 4, or 9 & 9 so that when 1 is added to each exponent the paired product = 100........for example the number generated from, 219 x 34. or from 29 x 39.

Equally 100 = 2 x 2 x 5 x 5 so another number with 100 divisors = 2 x 3 x 54 x 74 And similarly 100 = 4 x 5 x 5 and we can put the exponents in any order so another number with 100 divisors = 24 x 33 x 54. This last number = 270,000.