Question 190
Kate wants to write a set of distinct positive integers on the board, with each integer not exceeding $2023$, such that the sum of any two numbers written on the board is not divisible by their difference. For example, $4 \nmid 10$, $13 \nmid 27$, and $17 \nmid 23$, so Kate can write the numbers $3$, $7$, and $20$.
(1)
Show that Kate can't write $676$ numbers on the board.
(2)
Show that Kate can write $675$ numbers on the board.
(1)
Show that Kate can't write $676$ numbers on the board.
(2)
Show that Kate can write $675$ numbers on the board.