Question 16
Suppose $a,$ $b,$ $c$ are positive integers such that $$
a + b + c = 23
$$ and $$
\begin{aligned} &\gcd(a, b) + \gcd(b, c) \\ &\quad {}+ \gcd(c, a) = 9. \end{aligned}
$$ What is the sum of all possible distinct values of $a^2 + b^2 + c^2?$
a + b + c = 23
$$ and $$
\begin{aligned} &\gcd(a, b) + \gcd(b, c) \\ &\quad {}+ \gcd(c, a) = 9. \end{aligned}
$$ What is the sum of all possible distinct values of $a^2 + b^2 + c^2?$