Originally Posted by

**akhayoon** Let a1, a2....an be an arbitrary permutation of the numbers 1, 2,......, n,

where n is an odd number. Prove by contradiction that the product

(a1- 1)(a2 - 2)...(an - n) is even.

Hint: Try to use the fact that the sum of an odd number of odd integers is odd.

My Solution:

I know that I have to prove that [(a1- 1)(a2 - 2)...(an - n) is odd] is false to able to prove this by contradiction.

I don't understand how the hint can help me. but the only thing I know so far is that the product of odd numbers is odd (always). Not sure though how this can help me.