The product of 22 integers is equal to 1. Show that their sum cannot be 0.?

Answer 1

if the product is 1 this means that all of the numbers are 1 or -1.

to have a sum 0 you need equal number of 1s and -1s. the product of 11 1s and 11 -1s is -1 so the sum can't be 0.

Answer 2

Assume the product of two integers is equal to 1; that is

xy = 1 for some integers x and y.

To show that their sum cannot be 0, let's try a proof by contradiction.
Suppose their sum CAN be 0; that is,

x + y = 0

It then follows that
y = -x, so plugging this into our equation xy = 22, we have

x(-x) = 22

-x^2 = 22
x^2 = -22, which means
x = ± √(-22)

BUT, x is an integer, so this is a contradiction.

Therefore, the sum of two integers whose product is 22 cannot be 0.

Answer 3

The product of 22 integers is equal to 1.

The only way for 22 integers to equal 1 is if they are all either 1 or -1. (No other integers can be multiplied to get 1 as an answer.)

Since the product is positive, all the integers must be positive, or there must be an even number of negative integers.

In order for the sum to be 0, you would need an equal number of positive and negative integers. 22/2 = 11, so you would need 11 positive integers and 11 negative integers.

However, if you have 11 negative integers, then your product would be negative, not positive.

So, the sum of the integers CANNOT be 0.

Answer 4

well negative numbers count as integers too. and integers are whole numbers, so create a combo of positive and negative whole numbers that equal one.