How many ways to write one million as a product of three positive integers? In how many ways can the number 1;000;000 (one million) be written as the product
of three positive integers $a, b, c,$ where $a \le b \le c$?
(A) 139
(B) 196
(C) 219
(D) 784
(E) None of the above
This is my working out so far:
$1000000 = 10^{6} = 2^{6} \cdot 5^{6}$ and then to consider this as the product of three factors i.e.
$10^6 = 2^6 \cdot 5^6$
$= 2^a 5^p \cdot 2^b5^q \cdot 2^c 5^r$ (where $a+b+c = 6$ and $p+q+r = 6$).
However there are repetitions here because $2^3\,5^3 \cdot 2^2\,5^2 \cdot 2^1\,5^1$ is the product of the same three factors as $2^2\,5^2 \cdot 2^3\,5^3 \cdot 2^1\,5^1$.
I think there are 139 such factors.

As we can factorise one million as $5^6 \times 2^6 $, it becomes a matter of how many ways we can split up a set of size 12 into three disjoint subsets. This is more of a combinatronics problem, really.
