Here are some GCD questions I am struggling with:
1. Prove that, for a,n positive integers:
(a) n is divisible by gcd(a,a+n). (hint use the division identity M=aq+r)
(b) Hence show that GCD(a,a+1)=1
2. Show that for a,b gcd(a,b)=gcd(a,-b)=gcd(-a,b)=gcd(-a,-b)
3. prove that, for a,b,c,x, y integers tha c=ax+by if and only if gcd(a,b) divides c
4. Prove that for a,b,x,y integers if ax+by=gcd(a,b), then x and y are relatively prime.
5. Prove that the product of any three consecutive integers is divisible by 6.
Any help much appreciated.