Pick any positive integer you want, and look at its divisors (or factors).  For example, the divisors of 15 are 1, 3, 5, and 15; the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24;  and the divisors of 29 are 1 & 29.  Now disregard the number itself , and consider the other factors (these are called proper divisors, FYI.)  So the proper divisors of 15 are 1, 3, and 5.

Now, if you were to ADD those proper divisors, it should make sense that the SUM of these divisors could only be less than, equal to, or greater than the original number.  (In the example above, 1 + 3 + 5 = 9, which is less than the starting number 15.)  Let’s look at these three cases, in order of how rare they are:

1.  If a number’s proper divisors add up to the number itself, we call the number perfect (partly because this is SO rare!).  Example: 6 is perfect, since its proper divisors are 1, 2, & 3 and 1 + 2 + 3 = 6.

Interesting Facts about perfect numbers:

*The first 4 perfect numbers are 6, 28, 496, and 8128.  These 4 were known to the anient Greeks.

*There are currently only 48 known perfect numbers.  (Many of them are HUGE.)  All of them are even.  

*It can be proven that all (even) perfect numbers end in 6 or 8.

*No one knows if there are any odd perfect numbers.  (Find one, and make us both famous.  :-)).

2.  If a number’s proper divisors add up to less than the number, we call the number deficient. Example:  The proper divisors of 9 are 1 & 3.  Since 1 + 3 adds to less than 9, we call 9 deficient.

Interesting Facts about deficient numbers:

*All prime numbers are deficient.  (If you think about this, you can explain why.)

*In some sense, deficient numbers are ‘rarer’ than abundant numbers (see below).  They’re both ‘infinite’, but there are almost always fewer deficient numbers than abundant in a given stretch of 100 numbers.  

3.  If a number’s proper divisors add up to more than the number, we call the number abundant.  Example:  The proper divisors of 12 are 1,2,3,4, & 6.  Since they add up to 16, which is more than 12, we call 12 abundant.

Interesting Facts about abundant numbers:

*There are odd abundant numbers, but they are relatively rare.  The smallest odd abundant number is 945. (But it can be proven that there are an infinite number of odd perfect numbers!)

*If you have an abundant number, then 4 times that number will also be abundant.

Bonus for Teachers and Explorers

So, every positive integer, then, is either deficient, abundant, or perfect.  It is fun to explore, say, the first 50 (or more) positive integers and keep tally or make a graph of how many of each.  This is particularly fun (and instructive – think of the work with divisors!) for students of almost any level above 4th or 5th grade.  Try it out!!