Question

The number of positive integral solutions of ABC =30 is?

Answer 1

We first factorize `30` into primes.

Explanation:

`30=2xx3xx5`
These are exactly `3` prime factors, and the first solution

If we consider `1` to be a factor we have more solutions:
`30=1xx2xx3xx5` and we can take one of the primes to be the second factor, and the product of the other two to be the third:

`30=1xx2xx15`
`30=1xx3xx10`
`30=1xx5xx6`

And we have the very obvious:
`30=1xx1xx30`

For a total of `5` solutions

If the order of A, B and C is important (that is if `2,3,5` is different from `2,5,3`) then there are more solutions still:

The first four solutions can be done in six orders each, and the fifth solution can be done in three orders.

Total `27`