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/
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