Each of the numbers
is either 1 or -1.
If the sum
,
prove that
must be a multiple of .
If each of the numbers is either a 1 or a -1, then each of the products in the sum above must be 1 or -1. In order for the entire sum to be 0,
(1) the number of these products added must be even, meaning that n must be even, and
(2) half of these products (n/2) equal 1 and half of these products (n/2) equal -1.
Now, instead of taking the sum of these products, let's take the product of these products. This product would be
.
On the other hand, each of appears 4 times, so . This means that n/2 must be even, which means that n must be a multiple of 4.
01