This is actually my own question about GDC GDC Thm says that

This is actually my own question about GDC.

GDC Thm says that \"Let m1, m2 be positive integers. Let d be a positive generator for the ideal generated by m1, m2. Then d is a greatest common divisor of m1 and m2,\" which can be written as\" d=gcd(m1, m2) = am1+bm2 where a,b are positive integers. \"

But assume m1=3, m2=2, clearly, gcd(3,2) = 1, but there is no a and b can satisfy 1=a*3+b*2. Anyone can solve my question? Or i am wrong somewhere.

Solution

As you have already assumed that a, b are positive integers .You have taken m1= 3 and m2=3.

Lets say a=3 and b=2 and

So, am1 = 9 and bm2 = 4 .So, gcd( 9, 4) =1

So, there are possible solutions and you can find others also,