Amicable Numbers

By Chandrajeet K

The term “amicable” means “friendly”.
Amicable numbers are friendly numbers.

Amicable numbers are a pair of numbers such that the sum of the proper divisors of each number equals the other number in the pair.

Okay…

What are proper divisors?

Proper divisors are all the numbers that divide evenly into a number, including 1 but excluding the number itself.

Example

1, 2, 3, 4, 6, and, 12 are divisors of the number 12. They divide evenly into 12 without a reminder.
But…the PROPER divisors of 12 are: 1, 2, 3, 4, and 6; we should NOT include 12 here.



Let’s go back to “Amicable Numbers”.
220 and 284 are the smallest pair of amicable numbers.

The proper divisors of 220 are:
1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110

The sum of all these proper divisors is:
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

Wow! That’s the other number in the pair.

Let’s now find the proper divisors of 284.
1, 2, 4, 71, and 142

Add them up.
1 + 2 + 4 + 71 + 142 = 220

Excellent! Indeed they are friendly numbers!

Easier said than done… It’s a lot of work to find divisors of big numbers.
Here are a few tips to help you finding the divisors and sum of the divisors of a number.

# 1

The quickest way to find the divisors of a number could be to obtain the prime factors of the number and then combine those factors in all possible ways.

Let’s look at an easy example.

You can use a factor tree to find the prime factors.

The prime factors of 24 are: 2, 2, 2, and 3
Let’s combine the factors in all possible ways to find the divisors of 24.
There are three 2’s and one 3 in 24.
Let’s first combine the 2’s.

2 is divisor of 24
2 x 2 = 4 is a divisor of 24
2 x 2 x 2 = 8 is a divisor of 24

Let’s now combine the 2’s and the 3.

3 is a divisor of 24
2 x 3 = 6 is a divisor of 24
2 x 2 x 3 = 12 is a divisor of 24
2 x 2 x 2 x 3 = 24 is a divisor of 24

So, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
We include 1 among the divisors, because ‘1 is a divisor of every number’.
Leaving out the 24 we get the PROPER divisors of 24.
The next step is to find the sum of proper factors of 24 which is:

1 + 2 + 3 + 4 + 6 + 8 + 12 = 36

Phew, that’s tedious! There’s got to be a better way.

What if we can get the SUM OF THE PROPER DIVISORS straight away?

Well…there’s a formula available for the sum of the divisors. This might come in handy when you got to work with pairs of really big amicable numbers.

# 2

Let ‘n’ be a positive integer.
n = (p^a) x (q^b) x (r^c) x … represents the prime factorization of ‘n’.

Then the sum of ALL the divisors of ‘n’ is:
{[p^(a+1)] – 1}/(p – 1) x {[q^(b+1)] – 1}/(q – 1) x {[r^(c+1)] – 1}/(r – 1) x …

Let’s try using this formula with the SAME SIMPLE example (24).

The prime factors of 24 are: 2, 2, 2, and 3
The prime factorization of 24 is (2^3) x (3^1).

So, the sum of ALL the divisors of 24 is:

{[2^ (3+1)] – 1}/(2 – 1) x {[3^(1+1)] – 1}/(3 – 1)
= {[2^4] – 1}/(1) x {[3^2] – 1}/(2)
= {[16] – 1}/(1) x {[9] – 1}/(2)
= {15}/(1) x {8}/(2)
= 15 x 4
= 60

REMEMBER 60 is the sum of ALL the divisors of 24. But what we need is the sum of the PROPER divisors of 24. So, SUBTRACT 24 from 60, we get:
60 – 24 = 36

Hooray! It works! It matches with our answer!

Hope this helps!