1. ## divisibility proof

The question I am working on is
Prove that a/b if and only if a^3/b^3.

I know that if p/a^n then p/a...but how can i apply that here..can i make a prime on my own despite the question did not say?

2. Originally Posted by empressA
The question I am working on is
Prove that a/b if and only if a^3/b^3.

I know that if p/a^n then p/a...but how can i apply that here..can i make a prime on my own despite the question did not say?
I'm pretty sure we can go through with bi-implications this way.

$\displaystyle a|b \Leftrightarrow b = ka$ for some $\displaystyle k \in \mathbb Z$

$\displaystyle \Leftrightarrow b^3 = k^3a^3$

$\displaystyle \Leftrightarrow a^3|b^3$ since $\displaystyle k \in \mathbb Z \implies k^3 \in \mathbb Z$

and done.

Now if that makes you uncomfortable, and you want to do both directions, change all the above $\displaystyle \Leftrightarrow$'s to $\displaystyle \implies$'s and consider that your proof of the forward direction.

For the backwards direction, you will need to work a little harder. Personally, I'd do a proof by contradiction, employing the division algorithm.