Prove or find a counerexample a If ax by 2 and a is odd th

Prove or find a counerexample:

a) If ax + by = 2 and a is odd, then gcd(a, b) = 1.

b) If ax + by = 3 and a is even, then gcd(a, b) = 1.

Solution

a)

True

a is odd so 2 cases

Case 1. a=1

Then gcd(a,b)=1 for all b

Case 2:a>=3

Then since we have x,y so that :ax+by=2 so gcd(a,b)<=2

But, a is odd

So gcd(a,b) cannot be 2 hence it must be equal to 1

b)

False

a=6,b=9

gcd(a,b)=3

6*(-1)+9*1=3