$\gcd(a^2-ab+b^2, a+b)$

Number Theory Level 3

If $a$ and $b$ are two coprime positive integers, what is the sum of all possible values of the following expression?

$\large \gcd(a^2-ab+b^2, a+b)$

1 solution

Muhammad Rasel Parvej
Jan 27, 2018

$gcd(a^2-ab+b^2, a+b)$

$=gcd((a+b)^2-3ab, a+b)$

$=gcd(-3ab, a+b)$ [As $(a+b)^2$ is a multiple of $(a+b)$ ]

$=gcd(3ab, a+b)$

$=gcd(3,a+b)$ [As $a$ and $b$ are coprime]

So, possible values are─ $1$ and $3$ . Hence $3+1=\boxed{4}$ is the answer.