Divisor function problem.

Divisor function problem.

I have a question asking for patterns in numbers of factors.
We had to say how many factors are in 234 (12), 768 has 2^8*3
I should probably write that I get the pattern now and it holds true for
all numbers.
If I write a number in its prime factors
234 = 3^2*2^1*13^1
Basically 3 can appear 2 times, but there is the possibility of factoring
without using the number 3, so that means there are 2+1 times, similarly
for 2 has 1+1 times and 13 has 1+1 times
therefore (2+1)(1+1)(1+1) = 12 total factors and it works for all numbers?
9 = 3^2 = 2+1 factors...
Am I on the right track now? Thanks all!