Summing Factors

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:

Four perfect numbers have been known since antiquity — the 3rd and 4th being 496 and 8128. However the fifth perfect number was not discovered until the 15th century, and has the value 33 550 336. Did you notice how all these perfect numbers are even? To this day it is not known if there are any odd perfect numbers. It is not even known if there is a largest perfect number, or, like the primes, they go on forever. By 2013, only 48 have been found, with the largest having close to 35 million digits!

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?

Counting Factors

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?

  • Num factorsDescription (p and q are primes)
    2Only prime numbers have exactly 2 factors.
    3The factors must be 1, p and p2, so these numbers are all squares of primes.
    4The product of two primes pq has the 4 factors 1, p, q and pq, but also primes cubed have the 4 factors 1, p, p2, p3.
    5The 4th power of a prime has exactly the 5 factors 1, p, p2, p3, p4.
  • Consider the number an where a is prime. Using what you've learned above, how many factors will this number have?

    The factors are 1, a, a2, a3,… an, so there are a total of n + 1 factors.

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 anb where a and b are prime. Using what you've learned above, how many factors will this number have?

    From an we have n + 1 factors 1, a, a2, a3,… an,
    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 anb2 where a and b are prime.

    From an we again have factors 1, a, a2, a3,… an,
    but this time we have both b and b2 times each of these,
    so there are a total of (n + 1) × 3 factors.

  • How might this generalise to anbm?
  • What about anbmc ld 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.