I had two questions I can't figure out...
1) prove that if gcd(a,b) = 1 then gcd(a+b,a*b) = 1
2) let gcd(a,b) = 1. prove that d= gcd(a+b, a-b) = 1 or 2
(hint: prove d<2)
Thanks!
Let d=gcd(a+b,a*b)
Then d divides (a+b)x+(a*b)y, where x and y can be any integers.
If you let x=a and y=-1, you get : a²+ab-ab=a². Thus d divides a²
If you let x=b and y=-1, you get : ab+b²-ab=b². Thus d divides b²
---> d=1.
d divides (a+b) and (a-b) and thus divides (a+b)+(a-b)=2a and (a+b)-(a-b)=2b.2) let gcd(a,b) = 1. prove that d= gcd(a+b, a-b) = 1 or 2
(hint: prove d<2)
Thanks!
So d divides 2 or d divides a. If it doesn't divide 2, it divides a and b. Then...
If it divides 2, then...