And it didn't occur to you to tell us that to begin with? sigh.
If n= (a; b), the greatest common divisor of a and b, since n is a common divisor, then a= xn and b= yn. a+ b= xn+ yn= (x+y)n so that n is a common divisor of a and a+ b. Now, suppose there were a larger common divisor- that is there exist m such that a= mj, a+ b= mk for integers j and k and m> n. The b= (a+ b)- a= mk- mj= m(k- j) so that m is also a common divisor of b.