Find the number of distinct positive divisors of $2^23^25^3$.

Question

Find the number of distinct positive divisors of $2^23^25^3$.

My Attempt:

$(2+1)(2+1)(3+1)=36$

But the answer given is $72$.

Maybe the answer given is wrong?

Or maybe there is something more here, like taking some combinations to make some factors negative and the remaining positive? But I don't think we'll get any extra positive divisors this way.

Answer 1

Your answer is correct. One can simply list all distinct positive divisors of $2^\color{red}{2} \cdot 3^\color{red}{2} \cdot 5^\color{red}{3} = 4\,500$:

1, 2, 3, 4, 5, 6, 9, 10, 12,
15, 18, 20, 25, 30, 36, 45, 50, 60,
75, 90, 100, 125, 150, 180, 225, 250, 300,
375, 450, 500, 750, 900, 1125, 1500, 2250, 4500

There are $36$ members, which is exactly $(\color{red}{2} + 1) \cdot (\color{red}{2} + 1) \cdot (\color{red}{3} + 1)$.