Edited to add:
Prove that for every n, gcd(n+1, n^2-n+1) is either 1 or 3.
I'm having a lot of difficulty with some of these proofs. I'm not quite sure where to begin...
First, If (a,b)=d, then prove (a/d, b/d)=1 (where a/d is a fraction... I can't figure out how to get the cool math symbols in here like everyone else!)
So far I have: d/a (d divides a) and d/b, so a=dv and b=dw where v,w belong to the set of intergers. Then I can plug that in, so (a/d, b/d) becomes (v,w) but that doesn't really get me anywhere... help?
Also, If (a,c)=1 and (b,c)=1, prove that (ab,c)=1.
So far I have 1=av+cw for some v,w belonging to the set of intergers. Similarly, 1=bx+cz, but that's as far as I've been able to get.