. Suppose the positive integer n is odd. First Al writes the numbers $1, 2,..., 2n$ on the blackboard. Then he picks any two numbers a, b, erases them, and writes, instead, $|a − b|$. Prove that an odd number will remain at the end.
I have proved it in this way, please check if it's correct:-
Clearly, there will be n odds and n even numbers
possibilities of picking a and b integers:-
$2$ odds -> $|a-b|$= even number Hence, sum of the series ->$(n-2)$odds $+$ evens $=$ odd integer
when a & b are both even numbers -> sum = $n$ odds $+$ $(n-2)$ even $+$ even $=$ an odd integer
Without losing generality,
when a is odd and b is even -> sum = $(n-1)$odd $+$ $n$ even $+$ odd $=$ odd integer
Hence, the last integer left shall also be an odd int.
Is my proof correct, please check, if not then kindly give me a hint, not a full solution...since I am preparing for math olympiad
Thank you