Hi, I read definition of GCD theorem, from book and from mathWorld website. " There are two different statements, each separately known as the greatest common divisor theorem. This does not make sanse 1. Given positive integers and , it is possible to choose integers and such that , where is the greatest common divisor of and (Eynden 2001). This make sense 2. If and are relatively prime positive integers, then there exist positive integers and such that (Johnson 1965). " ====================================== if I take 2nd definition from above and take m=12 n=7 then divisors: 12X1,6x2,4x3 7x1 gdc=1 and according to second definition. x=3 y=5 xm - ny = 1 (12x3 ) - (7x5) = 1 36 -35 = 1 it make sense. ======================================== but 1st definition says , we take again m=12 and n=7 no matter what values of 'x' and 'y' we pick, we can not make d smaller so it can become '1'. Unless we select negative x and y.