The Proper Divisors of a number are all its factors except for the number itself. So, for example, the proper divisors of 6 are 1, 2 and 3 since 1 × 2 × 3 = 6. However, the Ancient Greeks noticed that 1 + 2 + 3 = 6 as well, and because of this decided to call 6 a Perfect Number — a number whose proper divisors sum to itself. Thus began the search for other "perfect" numbers.

- There is one other perfect number less than 100. Can you find it?
Use the following tool to help your search:

When the sum of the proper divisors is not equal to the original number, the number is either deficient, meaning the sum is less than the original number, or abundant, meaning the sum is greater. The Pythagorean mathematician Nicomachus, writing his Introduction to Arithmetic in 100AD, first introduced this way of classifying numbers: perfect, deficient or abundant. Using the colourful language typical of his time he writes:

In the case of the too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is
produced wanting, defaults, privations and insufficiencies. And in the case of those that are found between the too
much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and
things of that sort — of which the most exemplary form is that type of number which is called perfect.

He goes on to describe abundant numbers as like an animal:
with ten mouths, or nine lips …
or with a hundred arms

whereas deficient numbers are
like an animal: with a single eye, … one armed

Try using the divisors tool above to find some deficient and abundant numbers.

- Which of perfect, abundant or deficient do you think is the most common type?
- Do you think this might change as numbers become very large?
- What type of numbers are primes?

A number that has just 2 factors is a prime number, since those 2 factors must be 1 and itself.

- Can you work out what kind of numbers have exactly 3 factors?
- How about numbers with exactly 4 factors, or exactly 5 factors?
- Consider the number
*a*^{n}where*a*is prime. Using what you've learned above, how many factors will this number have?The factors are 1,

*a*,*a*^{2},*a*^{3},…*a*^{n}, so there are a total of*n*+ 1 factors.

Num factors | Description (p and q are primes) |
---|---|

2 | Only prime numbers have exactly 2 factors. |

3 | The factors must be 1, p and p^{2},
so these numbers are all squares of primes. |

4 | The product of two primes pq has the 4 factors 1,
p, q and pq,
but also primes cubed have the 4 factors 1, p,
p^{2}, p^{3}. |

5 | The 4th power of a prime has exactly the 5
factors 1, p, p^{2}, p^{3}, p^{4}. |

We also saw that *ab* has factors 1, *a*, *b* and *ab*.

- Consider the number
*abc*where*a, b*and*c*are distinct primes. How many factors will this number have?This has factors 1,

*a*,*b*,*c*,*ab*,*ac*,*bc*, and*abc*, so there are a total of 8 factors.

Let's try it and see, using the following tool to help.

Choose three values for *a, b* and *c* and test for factors.
For example 2 × 3 × 5 = 30 has factors 1, 2, 3, 5, 6, 10, 15 and 30 — i.e. 8 as expected.

We can combine these facts to generalise our factor counting.

- Consider the number
*a*^{n}*b*where*a*and*b*are prime. Using what you've learned above, how many factors will this number have?From

*a*^{n}we have*n*+ 1 factors 1,*a*,*a*^{2},*a*^{3},…*a*^{n},

but we also have*b*times each of these,

so there are a total of (*n*+ 1) × 2 factors. - Going one step further, we can count factors of the number
*a*^{n}*b*^{2}where*a*and*b*are prime.From

*a*^{n}we again have factors 1,*a*,*a*^{2},*a*^{3},…*a*^{n},

but this time we have both*b*and*b*^{2}times each of these,

so there are a total of (*n*+ 1) × 3 factors. - How might this generalise to
*a*^{n}*b*^{m}? - What about
*a*^{n}*b*^{m}*c*^{ l}*d*^{ k}…? - Would any of this reasoning work if our
*a*,*b**etc*were not prime?

Successful mathematical reasoning like the above illustrates the power of primes in understanding the deep underlying properties of numbers and arithmetic.