Prove that $(a_1 − 1)(a_2 − 2)...(a_9 − 9)$ is always an even number. $a_1, a_2,..., a_9$ is an arbitrary permutation of positive integers from 1 to 9. Prove that $(a_1 − 1)(a_2 − 2)...(a_9 − 9)$ is always an even number.
So I don't understand what the question is asking by "arbitrary permutation of positive integers". Does that mean consecutive integers or multiples? And also, can someone please help me prove this too? Thank you so much! :)

The only way the product $(a_1 − 1)(a_2 − 2) \dots (a_9 − 9)$ could be odd is if $a_1, a_3, a_5, a_7$, and $a_9$ are all even numbers. That would take $5$ even numbers ...
