Problem 4

A keypad is a $3$ by $3$ grid with the digits $1$ to $3$ in order on the top row, $4$ to $6$ in order on the second row and $7$ to $9$ in order on the third row.

Fred has the keypad upside-down (that is, rotated through $180^\circ$) and types his four-digit PIN as if the keypad were the right way up. The number he types in turns out to be $2025$ more than $k$ times his original PIN where $k$ is a positive integer. Prove there is only one possible PIN and find it.