Products of Integers (Even or Odd)

Date: 03/23/2002 at 14:48:31
From: Kristine Hodges
Subject: Product of two even integers proof

How can I prove that the product of two even integers is an even 
integer? And that the product of two odd integers is an odd integer?  
I know that it is true, but I don't know how to put it in an equation.

Thanks.
Kristine Hodges

Date: 03/23/2002 at 15:13:23
From: Doctor Paul
Subject: Re: Product of two even integers proof

Let x and y be even numbers. Then x = 2*a and y = 2*b for some 
integers a and b.

Then x*y = 2*(2*a*b), which is even.

Now let x and y be two odd numbers. Any odd number is always one more 
than some even number. Thus we can write x = 2*a + 1 and y = 2*b + 1 
for some integers a and b.

Then 

   x*y = (2*a + 1) * (2*b + 1) 
       = 4*a*b + 2*a + 2*b + 1 
       = 2(2*a*b + a + b) + 1, 

which is one more than an even number and is hence odd.

I hope this helps.  Please write back if you'd like to talk about this 
some more.

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