These are straight forward ( ). If then so . However, so which means . Likewise for all the other ones.2. Show that for a,b gcd(a,b)=gcd(a,-b)=gcd(-a,b)=gcd(-a,-b)
If let then LHS is divisible by so . Now for conserve. We can write since it means and so .3. prove that, for a,b,c,x, y integers tha c=ax+by if and only if gcd(a,b) divides c
You can write , now it is clear that .4. Prove that for a,b,x,y integers if ax+by=gcd(a,b), then x and y are relatively prime.
We want to prove . Notice that and . Thus, it is sufficient to prove that . It is easy to see why that is true.5. Prove that the product of any three consecutive integers is divisible by 6.