Adding Odd and Even Numbers

Date: 05/28/2003 at 16:16:23
From: Sherry
Subject: Odd and even numbers

Can you add any two numbers (one odd and one even) to total an even 
number?

Can you add any two odd numbers to total an odd number?

It's simple NO, right??

Date: 05/28/2003 at 20:55:47
From: Doctor Achilles
Subject: Re: Odd and even numbers

Hi Sherry,

Thanks for writing to Dr. Math.

You are absolutely correct in both cases. Here's a semi-formal 
explanation why:

Let's simplify numbers and just use tallies. We write each number as 
a set of lines. The number of lines represents the value of the 
number.

So, for example, the number 1 would be:

  |

The number 3 would be:

  |||

The number 10 would be:

  ||||||||||

Etc.

For any number, you can group the lines into pairs (sets of 2).  So 
you could write 10 as:

  || || || || ||

And you could write 11 as:

  || || || || || |

Notice that if we do that, every even number will pair up perfectly, 
whereas odd numbers will always have an odd-line-out.

If we add two even numbers, then both of the numbers we start with 
will have a partner for every line, so the sum will have a partner 
for every line.

If we add two odd numbers, then both of the numbers will have one odd-
line-out; the two odd-lines-out can be paired, resulting in each line 
having a partner (therefore the sum will be even).

If we add an odd and an even, then the odd-line-out in the odd number 
will not find a partner in the even number (because all the lines in 
the even number will be taken) so there will still be an odd-line-out 
in the sum, so the sum will be odd.

It doesn't matter which odd or even numbers you pick, all that matters 
is whether there were any odd-lines-out to start with and whether they 
found other odd-lines-out to pair up with or were left still alone at 
the end.

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