## ashur 3 years ago How do I prove that every powerful number can be written as the product of a perfect square and a perfect cube?

1. ilikephysics2

2. ashur

they are even

3. ilikephysics2

there you go thats all it is

4. KingGeorge

Just curious, but what class is this for?

5. asnaseer

how does that prove it? I didn't know what "powerful numbers" were until I just looked them up. if I understand it correctly then a powerful number is a positive integer m such that for every prime number p dividing m, p^2 also divides m.

6. ashur

Proofs class

7. ilikephysics2

I didn't really understand the question

8. asnaseer

:/

9. KingGeorge

I was curious because I helped out on the same question yesterday. See http://openstudy.com/study#/updates/50552fcde4b02986d370aedd

10. ashur

The first part makes sense but why are u subtracting 3 form ei when ei is odd?

11. KingGeorge

So that I get $\large p_i^{e_i}=p_i^{f_i+3}=p_i^{f_i}\cdot p_i^3$Note that since $$e_i$$ is odd, $$e_i-3$$ is even, so $$\displaystyle p_i^{e_i-3}=p_i^{f_i}$$ is a perfect square.

12. KingGeorge

Additionally, $$p_i^3$$ is a perfect cube.

13. ashur

oh that makes perfect sense, thanks alot

14. KingGeorge

You're welcome.