Quandaries
and Queries
Hi MathCentral
I have this one proof that I almost get but am not quite how to go about
doing it .
I don't know how to ask this question.
QUESTION:
Let a and b integer and relatively prime. Proof that:
GCD (a + b , a2 - ab + b2) = 1 or 3
regards
Carol
Hi Carol,
Suppose that d > 1 and d divides a + b and a2 - ab + b2. Since gcd(a,b) = 1, either d does not divide a or d does not divide b. Suppose that d does not divide a.
Write
a2 - ab + b2 = (a + b)2 -3ab
Since d divides a + b, d divides 3ab. If d divides b then, since d divides a + b, d divides a. Thus d does not divide a or b so d divides 3. Thus d = 3.
Penny