Could someone help with the third question posted above please? Thanks.
Need proofs for the following lemmas:
- Let a, b, c belong to Z (integers) and suppose gcd(c,a)=1. If c|ab implies c|b.
- Let a,b,q,r belong to Z with a=qb+r, then
3. Let x,y,z belong to Z. Then gcd (x,y,z)=1 IF AND ONLY IF gcd(x,y) and gcd (x,z)=1
- gcd(a,b)=gcd(b,r)
- gcd(a,b)=gcd(-a,b)=gcd(a,-b)=gcd(-a,-b)
#1. If , then because are co-prime. Try to finish it from here.
#2 Define two sets and . These sets are nonempty since you can take . Now show .
Let . Then and . There are so that and . Substitute into your equation and you get . So and
Do the other direction now. Then . By definition,
The second part is very easy. Just note that if , then . How does that affect the gcd?