Question 166
The numbers $2$, $3$, $5$, $\ldots$, $2003$ are written on the board (here all prime numbers from the segment $[2; 2007]$ are listed). By a *simplification* operation we mean replacing two numbers $a$, $b$ by the largest prime number not exceeding $\sqrt{a^2 - ab + b^2}$. At the beginning a student removes the number $q$, $2 \lt q \lt 2003$, and then repeats the simplification operation as long as the board contains at least two numbers. Find the maximal and the minimal possible values of the number he obtains at the end. How do these values depend on $q$?