Question 66
A set $C$ of positive integers is called good if for any integer $k$ there exist $a, b \in C, a \neq b$, such that the numbers $a+k$ and $b+k$ are not coprime. Prove that if the sum of the elements of a good set $C$ equals $2003$, then there exists $c \in C$ for which the set $C \setminus \{c\}$ is good.