Division Quotient

Date: 11/06/2002 at 22:19:12
From: Lillian 
Subject: Division quotient

In a division sentence, when will the quotient be odd? When will it 
be even?

I've confirmed that if both numbers are even, the answer will be even, 
and if both numbers are odd, the answer will be odd. But when one 
number is even and the other odd, the answer could be either. Help.

Date: 11/07/2002 at 09:32:52
From: Doctor Peterson
Subject: Re: Division quotient

Hi, Lillian.

I assume you are talking about a division sentence in which all the 
numbers are whole, like 6/3 = 2. This will be equivalent to a 
multiplication, like 6 = 3*2. You can make a table for multiplication:

      * | even | odd         examples
    ----+------+-----
    even| even | even        4*6=24          4*5=20
    odd | even | odd         5*4=20          3*5=15

When we turn this inside out to make the corresponding table for 
division, we find that some spaces are empty: some divisions will not 
give a whole number, so the quotient will be neither odd nor even. 
Moreover, there are some cases where the quotient can be either even 
or odd. This is perfectly valid. Here is my result:

      / | even | odd         examples
    ----+------+-----
    even|  ?   | even        24/4=6, 20/4=5  20/5=4
    odd |  -   | odd         (5/2=2.5)       15/3=5

Note that you are wrong in saying that even divided by even is even, 
and also that the result is indeterminate if the numbers have 
different parity. Can you explain how you determined this?

So what is the answer to the question "when will the quotient be 
odd"? It CAN be odd when both numbers are even or both are odd; in 
the case where both are even, you need to look for something beyond 
the evenness of the numbers. I suggest thinking about the prime 
factorizations of the numbers.

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