Multiples of Four

Date: 03/01/99 at 02:09:10
From: felicity
Subject: Multiples of Four

My mathematics homework sheet says that 0 is a multiple of four, but 
I cannot see how as four does not go into 0 but it goes into the rest: 
4 8 12. I think multiples of four start with four. All the examples on 
my page start with 0. What do you think?

Thanks.

Date: 03/01/99 at 05:26:43
From: Doctor Reno
Subject: Re: Multiples of Four

I have looked through definitions of multiples on the Web and in 
textbooks here at home, and even in a dictionary! I think I have an 
explanation for the confusion. 

According to all the definitions I found, a multiple is a quantity 
into which another can be divided with zero remainder. Therefore, the 
multiples of four *would* be 0, 4, 8, 12, etc. (4 *does* go into zero: 
0 / 4 = 0).

However, no reference I found listed 0 as a multiple of any number, 
and I think this is because it isn't useful to us. If we need to know 
the least common multiple of 3 and 4, we would list them like this:

multiples of 3: 3, 6, 9, 12, 15...
multiples of 4: 4, 8, 12, 16...

and we can quickly see that 12 is the least common multiple. But if we 
include 0 on these lists, then 0 would *always* be the least common 
multiple of any pair of numbers. And that would not help us when we 
needed to use the least common multiple in order to find the least 
common denominator of two or more fractions. 

Zero can never be the denominator of a fraction, and that is probably 
another important reason why textbooks do not usually include it as a 
multiple of any number. 

So, I think that by definition, zero could be considered a multiple of 
4, but since it is of no use to us we don't include it in a list of 
multiples. Your textbook is an exception. Have you asked your teacher 
about this? I would be interested in what your teacher or others have 
to say about it!

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