Abundant and Deficient Numbers

Date: 10/14/97 at 12:28:17
From: Krysti Byers
Subject: Abundant and deficient numbers

I need information on what these are.  I also need some information on 
how these are used in elementary school teaching. For example, what 
are they used for?

Thank you!

Date: 10/14/97 at 18:37:36
From: Doctor Chita
Subject: Re: Abundant and deficient numbers

Hi Krysti:

Before defining deficient and abundant with respect to numbers, we 
first need to recall that the natural numbers are either prime or 
composite. Prime numbers are those numbers that have two unique 
factors: {2, 3, 5, 7, ... }. Composite numbers are made up of 
combinations of prime factors. The set of prime factors, or divisors 
of a number, determines whether a composite number is deficient, 
abundant, or perfect.

A deficient number is a composite number in which the sum of its 
factors is less than the given number. For example, the number 8 has 
factors (divisors) of 1, 2, 4, and 8. If you disregard 8 as a factor, 
then the sum 1 + 2 + 4 = 7 and 7 is less than 8. Therefore, 8 is 
deficient.

An abundant number is a composite number whose factors, without the 
number itself, have a sum greater than the number. For example, 12 has 
factors of 1, 2, 3, 4, 6, and 12. The sum 1+2+3+4+6 = 16, and 16 > 12.

A perfect number is one whose factors are equal to a given number. So, 
6 is perfect because 1+2+3 = 6. 

Identifying these interesting numbers provides motivation for students 
to develop their number sense while practicing their arithmetic facts. 
You could have students make a table listing the numbers from 1 to 100 
in Column 1. In Column 2, have them write all the factors of each 
number, including 1 and the number. In Column 3, have them find the 
sum of the factors (without including the number itself). In Column 4, 
identify the nature of the number. 

Have students work together to draw conclusions from their table. What 
patterns do they see? How many perfect numbers are there less than 
100? Are there more abundant or deficient numbers? Are square numbers 
perfect? What about even numbers? Odd numbers?

There are many other questions you can have students think about as 
they explore the numbers. As they answer questions, they come to new 
understandings about numbers. There is also a lovely little paperback 
book called _A Number for Your Thoughts_ written by Stephen P. 
Richards and published in 1982. If you can get hold of it, in it you 
will find abundant, if not perfect, stories about numbers. Have fun!

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/