Product Always an Even Number?

Date: 03/17/2002 at 09:46:05
From: Tian
Subject: Algebraic equation

Doctor Math,

The letters a1, a2, a3, a4, a5, a6, a7 represent seven positive whole 
numbers. The letters b1, b2, b3, b4, b5, b6, b7 represent the same 
numbers but in a different order. Will the value of the product 

   (a1 - b1)(a2 - b2)(a3 - b3)(a4 - b4)(a5 - b5)(a6 - b6)(a7 - b7) 

always be an even number?

Date: 03/17/2002 at 10:07:35
From: Doctor Jubal
Subject: Re: Algebraic equation

Hi Tian,

Thanks for writing Dr. Math.

Let's look at it this way. We want to know whether a product of 
several numbers is even or odd.

The product of two even numbers is even.
The product of an even and an odd number is even.
The product of two odd numbers is odd.

Therefore, the product of the seven differences (a1 - b1)...(a7 - b7) 
will be odd only if all seven differences are odd.

So, we need to know whether these differences are even or odd.

The difference of two even numbers is even.
The difference of two odd numbers is even.
The difference of an even and an odd number is odd.

Therefore, in order for the final product to be odd, every one of the 
differences (a1-b1)...(a7-b7) has to be the difference of an even and 
an odd number.

Can you tell me why this is impossible?

Write back if you'd like to talk about this some more, or if you have 
any other questions.

- Doctor Jubal, The Math Forum
  http://mathforum.org/dr.math/   

Date: 03/17/2002 at 10:54:26
From: Tian
Subject: Algebraic equation

Dr. Math,

I have figured out that the difference between the 2 numbers has to 
be odd because the product of 7 odd numbers will be odd. Can you 
please tell me why it is impossible or possible.

Thanks.

Date: 03/17/2002 at 12:52:32
From: Doctor Jubal
Subject: Re: Algebraic equation

Hi Tian,

In order for the product to be odd, all seven differences must be odd.  
In order for all seven differences to be odd, each one must be the 
difference between an odd number and an even one.  This requires seven 
even and seven odd numbers.

However, the numbers a1...a7 are the same numbers as b1...b7, just in 
a different order. That means each number in the differences appears 
twice. But if we have seven odd and seven odd numbers, there's no way 
each one could appear twice, because seven is an odd number. 
Therefore, at least one of the difference must contain either two even 
or two odd numbers, and so be even.

If there's even one even number in the product, it will be even.  
Therefore, it is impossible for the product to be odd.

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Jubal, The Math Forum
  http://mathforum.org/dr.math/