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### Products of Integers (Even or Odd)

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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.

some more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Number Theory