Rearrange and multiply

Number Theory Level pending

Let $n$ be an odd integer greater than 1.

Let $A(1)$ , $A(2)$ ... $A(n)$ be a rearrangement of the set { $1$ , $2$ , .... $n$ }

Consider the product $(A(1) - 1) \times (A(2) - 2) \times ... (A(n)-n)$

This product is...

Even Odd It can be either

1 solution

Denton Young
Aug 20, 2017

Since $n$ is odd, let $n = 2k +1$ , where $k$ is a positive integer.

Then the set $A(1)$ ... $A(n)$ contains $k$ even numbers and $k+1$ odd numbers.

For the product to not be even, every term has to be odd. But since there are $k+1$ odd numbers in the set of $A(i)$ 's and only $k$ even numbers, at least one odd $A(i)$ has to have an odd number subtracted from it, making the term even and hence the entire product an even number.

Rearrange and multiply

1 solution

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