How many ways to write one million as a product of three positive integers?

Question

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.

Answer 1

Hint: Solve the problem for $abc = 10^6$. This has $ {8 \choose 2}^2=784$ solutions.

Count the number of solutions where $a=b=c$.

Count the number of solutions where $a=b$ or $b=c$ or $c=a$.

Count the number of solutions where $a, b, c$ are pairwise distinct.

Account for your repeated counting above, to find the cases where $a \leq b \leq c$.

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Motivation: Bars and stripes, Orbit Stabilizier Theorem.

Answer 2

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.