[ Processes and Operations ] $[$ Invariants $]$

On the board, the numbers 1, 2, 3, ... 1984, 1985 are written. It is allowed to erase any two numbers and write down the absolute value of their difference instead. In the end, one number will remain on the board. Can it be zero?
For the given operations, the parity of the sum of all numbers written on the board does not change.

## Solution

The modulus of the difference of two integers has the same parity as the sum of these numbers. Therefore, the parity of the sum of all numbers does not change with the specified replacement. The sum of integers from 1 to 1985 is odd (there is an odd number of odd numbers among them). Therefore, it cannot become zero.

## Answer

It cannot.