I have such task:
Demonstrate, that, (a;b)=(a;a+b)
There is no more explanation. Teacher said that there is somthing to do with number sharing properties.
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.