Proof problem about gcd

Question

Prove the following statements:

1. gcd(a,b) = gcd(a-b,b)
2. if gcd(a,b)=d, then exists such x, y ∈ Z, that ax+by=d

Answer 1

Try a specific example

2 a= 27, b= 15. gcd(27,15) = 3 27x+15y= 3 means x and y are fractions, not integers so not elements of Z

one counterexample disproves a supposed identity

maybe you intended to prove lcm(a,b)=d? lcm(27,15) = 135

27x+15y=135 if x=5 and y=0 both integers or x=0 and y=9