Most people, when asked how many divisors the number 60 has (including 1 and 60), would struggle to do so without listing them all. Yet once you know that the prime factorisation of 60 is you can immediately say that the number of divisors is . In other words, you take each index, add 1 then multiply them together.
This is easy to see if you list the divisors as
It’s then not difficult to prove the general result:
- If then the number of divisors of is