Question

2. The G.C.D of (x+y) and x-y is

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Answer 1

Explanation:

Show that if gcd(x,y)=1 then gcd(x−y,x+y) is either 1 or 2

I think the question is asking me to show if the gcd(x,y) then the gcd(x−y,x+y) is 1 or 2.

so, for the first bit. Let d=gcd(a,b)d=gcd(a,b); by definition there are integers a′a' and b′b' such that a=a′da=a'd and b=b′db=b'd, so a′dx+b′dy=da'dx+b'dy=d. Dividing through by dd, then a′x+b′y=1'x+b'y=1.

Let e=gcd(x,y)e=gcd(x,y). As before, there are integers x′x' and y′y' such that x=ex′x=ex' and y=ey′y=ey'. Substituting these into the previous equation, we get a′ex′+′ey′=1a'ex'+'ey'=1, or e(a′x′+b′y′)=1e(a'x'+b'y')=1. Since a′x′+b′y′a'x'+b'y' is an integer, this implies that e=1 or e=−1: these are the only divisors of 11. But e is a greatest common divisor and hence by definition positive, so e=1