1. Divisibility

Prove that if $\displaystyle q$ does not divide $\displaystyle p$, then $\displaystyle q^n$ does not divide $\displaystyle p^n$.

This isn't actually homework or anything, just a hypothesis I observed.

2. Assuming that $\displaystyle p \& q$ are positive integers, any prime factor of $\displaystyle q^n$ must also be a prime factor of $\displaystyle q$. Can you use that to finish your question?

3. Originally Posted by DivideBy0
Prove that if $\displaystyle q$ does not divide $\displaystyle p$, then $\displaystyle q^n$ does not divide $\displaystyle p^n$.

This isn't actually homework or anything, just a hypothesis I observed.
Here is a little bad proof, but it makes sense.

$\displaystyle \frac{p^n}{q^n} = \left(\frac{p}{q}\right)^n$

Now, $\displaystyle \frac{p}{q}$ is not a whole number. So raised to any number it is not a whole number.

4. Originally Posted by ThePerfectHacker
Here is a little bad proof, but it makes sense.

$\displaystyle \frac{p^n}{q^n} = \left(\frac{p}{q}\right)^n$

Now, $\displaystyle \frac{p}{q}$ is not a whole number. So raised to any number it is not a whole number.
Hahaha, awesome!

5. Originally Posted by DivideBy0
Hahaha, awesome!
Ahhhh... But what TPH didn't tell you is that you have to be careful of the following trap:
$\displaystyle \sqrt{2}$ is not a whole number.

But $\displaystyle (\sqrt{2})^2 = 2$ is.

What you would need to show is something like that $\displaystyle \sqrt{2}$ cannot be expressed as a rational number $\displaystyle \frac{p}{q}$.

-Dan