How many triples of positive integers (a,b,c) satisfy:
$$\ a \le b \le c $$ when
$$\ abc = 1,000,000,000$$

Question

anonymous · Answer

Trial and error kind of thing?

shamil98 · Answer

It's combinatorics. I have no idea how to go about this one.

anonymous · Answer

Wow. Cool problem! I'll have to think about this. Hold on.

anonymous · Answer

To start, the prime factorization of our number is 2^9 * 5^9
All of our various possibilities will have those 2's and 5's assigned in various ways to a, b and c. Note that it's ok for none of these to be assigned to one two of the variables as we can just make that variable be 1.

anonymous · Answer

Prime factorization is important here.

anonymous · Answer

Since $$
10^9 = 2^95^9
$$Then it is just a matter of dividing up $2$s and $5$s among $a$, $b$, and $c$.  Normally any partition we created could be reused by swapping one variable for another, but since $a\leq b\leq c$, we must sort our partitions so order does not matter.

shamil98 · Answer

Hm, so how long do you think it would take to figure out all the possible combinations? Are there any other methods?

anonymous · Answer

There are smart ways of calculating the partitions.