Need help with this problem

Question

The question is as follows:

let there be a set a1,a2,a3,...an.

If n is odd, prove that if (a1-1)(a2-2)...(an-n)  **≠**  0, then the product is an even number.

From what I see, only odd number \* odd number gives an odd number, otherwise it is even. So why can't the above question be odd? Can't all the subtractions yield all odd numbers? Any help would be appreciated.

Answer 1

As stated, this is clearly false, since you can just take n = 1 and let a1 be 4, so the product is 4-1=3. Is there some part in the setup you missed?

Answer 2

Unless there is an additional restriction on set A = {a1…an} this is false. Let n = 3, A = {2, 3, 4}. (2 - 1) (3 - 2) (4 - 3) = 1

Answer 3

You’ve missed an assumption to this classical problem: (a1,…,an) is a permutation of {1,…,n}.

For a hint, consider the parity of each factor and the sum (a1-1)+…+(an-n)