## 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? 2 years ago 2 years ago

1. ilikephysics2

Think about it, what do you know about a square?

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.