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.
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:
Try using the divisors tool above to find some deficient and abundant numbers.
A number that has just 2 factors is a prime number, since those 2 factors must be 1 and itself.
|Num factors||Description (p and q are primes)|
|2||Only prime numbers have exactly 2 factors.|
|3||The factors must be 1, p and p2, 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, p2, p3.|
|5||The 4th power of a prime has exactly the 5 factors 1, p, p2, p3, p4.|
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.
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.
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.
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.
Successful mathematical reasoning like the above illustrates the power of primes in understanding the deep underlying properties of numbers and arithmetic.