Question 168
Suppose positive integers $a$, $b$, $c$ satisfy the following four conditions:
$\bullet$ $\text{gcd}\{a, b, c\} = 1$.
$\bullet$ $\text{gcd}(a, b + c) \gt 1$.
$\bullet$ $\text{gcd}(b, c + a) \gt 1$.
$\bullet$ $\text{gcd}(c, a + b) \gt 1$.
Here we denote by $\text{gcd}\{ \}$ the greatest common divisor of the numbers in $\{ \}$.
Determine the minimum possible value the sum $a + b + c$ can take.
$\bullet$ $\text{gcd}\{a, b, c\} = 1$.
$\bullet$ $\text{gcd}(a, b + c) \gt 1$.
$\bullet$ $\text{gcd}(b, c + a) \gt 1$.
$\bullet$ $\text{gcd}(c, a + b) \gt 1$.
Here we denote by $\text{gcd}\{ \}$ the greatest common divisor of the numbers in $\{ \}$.
Determine the minimum possible value the sum $a + b + c$ can take.