Why Two Odds Always Sum to an Even

Date: 03/21/2002 at 23:46:45
From: Will Dwyer
Subject: 2 odds always = even

My six-year-old son asked me why two odd numbers always equal an even 
number, but two even numbers never equal an odd number.

I have no idea. Initially I thought that because two odd numbers are 
each only a digit away from being even, the product of two such digits 
takes the addition of the two odd numbers up to an even number, but 
that seems a little too simple.

Any ideas?

Date: 03/22/2002 at 00:26:11
From: Doctor Jeremiah
Subject: Re: 2 odds always = even

Hi Will,

That's exactly what is going on. The odd numbers are one away from the 
even number. The even numbers add to an even number and the "one away"
bits add to 2 (which is also even).

Here is the mathematical way to say the same thing:

An even number is a multiple of two. So all even numbers have the form 
2k or 2j (some number j or k multiplied by 2).

If you add two even numbers:
  2j  +  2k
  2(j+k)
you still have an even number.

An odd number is an even number plus one. So an odd number can be 
represented with 2k+1 or 2j+1 (some number j or k multiplied by 2 
plus 1).

If you add two such numbers:
  2j+1   +  2k+1
  2j+2k  +  1+1
  2(j+k) +  2
the left bit is the even part;
the 1+1 is the sum of the odd parts;
but 1+1 always equals 2 so it can never become an odd number.

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

Date: 03/22/2002 at 00:27:46
From: Doctor Twe
Subject: Re: 2 odds always = even

Hi Will - thanks for writing to Dr. Math.

This is one way you can think of it: an even number of objects can 
be "paired." For example 10 apples can be matched up into 5 pairs of 
apples. An odd number of objects will have one left over when pairing. 
For example, 5 oranges can make 2 pair with 1 left over. (This is the 
definition of odd numbers, having a remainder of 1 when dividing by 
2).

With two odd numbers, each odd number by itself has one left over, but 
when we add them together, we can combine these two "leftovers" to 
form another pair.

Any time we add an even number of odd numbers, we'll be able to pair 
up the "leftovers" and get an even number for our sum. But if we add 
an odd number of odd numbers, we'll get an odd number for a sum. This 
is because we can only pair up 2 of the 3 leftovers (or 4 of 5, 6 of 
7, etc.)

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.com/dr.math/