Let a and b be distinct integers such that . Show that
Table Of Content
1. Show that
This is a tricky problem. The key here is that we can re-express the problem via coprime terms to gain some insight.
Express as multiples of coprime naturals
Let , then it must be the case that there exists a coprime pair such that . We know that , let’s see what it divides to
Here, we will employ a natural property of where to show that it must be the case that . In order to claim this, we only need to show that
Using the simple euclidean algorithm, we reduce the problem to
Because by definition , then naturally .
In the same vein, this reduces down to .
Now, because , it must also be the case that . Together with the lemma stated above, this concludes our proof that .
To the finish line!
Because , it must be the case that . Then, we can derive the following expression from the problem. Ordering such that so that
This concludes our proof that .