Question 12
Morgane writes the integers $3$, $4$, and $12$ on a board. She then repeatedly makes the following change: she chooses two numbers $a,b$ on the board, erases them, and replaces them by $(3a+4b)/5$ and $(4a-3b)/5$.
After finitely many such changes, can Morgane
a) write the number $13$ on the board?
b) write the number $-12$ on the board?
After finitely many such changes, can Morgane
a) write the number $13$ on the board?
b) write the number $-12$ on the board?