Find how many integer solutions are there to $x \cdot y\cdot z=12^5$?

Question

The first question asks for positive integer solutions to $x\cdot y \cdot z=5^9$.

I solved it by finding the coefficient of $x^9$ of the generating function $(1 + x^1 + x^2 + x^3 ... + x^9)^3$ since $5$ is a prime.

which gives the answer $28$

But the second question which asks for positive integer solutions to $x \cdot y\cdot z=12^5$

where 12 is not a prime.

Not sure my solving step is correct or not.

If it is not wrong then how do I deal the situation like the second question?

Any assistance will be appreciated. Thanks

Your generator polynomial should be $(1+x^1 + x^2 + x^3 ... + x^9)^3$ because power $0$ is possible (we get $1$'s). — Henno Brandsma, Jun 29 '19 at 07:30
Because the question ask for positive integer solution so I did't add 1 in it. — Tverous, Jun 29 '19 at 07:35
But 1 is equal to $x^0$ . Once you choose $x^0$ to be one of the combinations, you will set one of x, y, z to be zero? — Tverous, Jun 29 '19 at 07:42
no to $1$. $x=1,y=1,z=5^9$ is a solution to the original equation corresponding to coefficients $(0,0,9)$ in the product. — Henno Brandsma, Jun 29 '19 at 07:45

score 2 · Accepted Answer · edited Jun 29 '19 at 10:49

2

Write $12^5=2^{10}3^5$ and consider the two equations $xyz=2^{10}$ and $xyz=3^5$ using the method you used for the first question. The number of solutions to the original second question is the product of the numbers of solutions to these two sub-questions: $\binom{12}2\binom72=1386$.

edited Jun 29 '19 at 10:49

Tverous

51

answered Jun 29 '19 at 06:44

Parcly Taxel

103,344

Thanks for the help. – Tverous Jun 29 '19 at 07:17
@"Parcly Taxel", thanks to the help from @"Henno Brandsma"'s comments above. I realized that $x^0$ is acceptable for the generating function. So the answer should remain the same. Which means the answer you wrote in the first place, $\binom{12}2\binom{7}2$ ,is correct. Sorry. – Tverous Jun 29 '19 at 10:35

Find how many integer solutions are there to $x \cdot y\cdot z=12^5$?

1 Answers1