Prove: p - q Divides p - 1 Implies q - p Divides q - 1

[Dustinsfl]

Let p,q ∈ ℤ+ , p < q. Prove that if p - q divides p - 1, then q - p divides q - 1.

So if p - q divides p - 1, then k*(p - q) = p - 1.

Now what?

 

[snipez90]

Now you should relate the conclusion you want to the hypothesis. How can you rewrite q -1 to introduce the p - 1 term (and why would this help you).

 

[VeeEight]

Multiply through by k and then bring over the p from the right side (and simplify that with kp). Then subtract both sides by q and rearrange.

 

[Dustinsfl]

By doing that, it only yields k·p - k·q - p = p·(k - 1) - k·q ⇒ p·(k - 1) = k·q - 1. How can that be manipulated to fit the conclusion?

 

[VeeEight]

What I was thinking was: p·(k - 1) - k·q = -1, then add q to both sides to give q-1 on the right side and simplify -kq + q. Then rearrange the resulting left side of the equation.