suppose gdc(x,y)=1. What can gdc(x+y,x-y) be?

Answer 1

Do you mean gcd? Greatest Common Denominator? If so, then the answer is that it could be one or two..

How is this proved? Assume that there is a number Z that divides (X+Y) and also divides (X-Y). Then it must divide the sum as well. But the sum is 2X. By the same logic, it must divide the difference. But the difference is 2Y. Therefore, Z must divide both 2X and 2Y. That means it is either 2 or it is a number that must divide Z and Y -- but we know that only 1 divides X and Y.