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

1. ilikephysics2 Group Title

2. ashur Group Title

they are even

3. ilikephysics2 Group Title

there you go thats all it is

4. KingGeorge Group Title

Just curious, but what class is this for?

5. asnaseer Group Title

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 Group Title

Proofs class

7. ilikephysics2 Group Title

I didn't really understand the question

8. asnaseer Group Title

:/

9. KingGeorge Group Title

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

10. ashur Group Title

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

11. KingGeorge Group Title

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 Group Title

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

13. ashur Group Title

oh that makes perfect sense, thanks alot

14. KingGeorge Group Title

You're welcome.