Amicable Numbers

The term“amicable” means “friendly”.

Amicablenumbers are friendly numbers.

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

Okay…

What areproper divisors?

Properdivisors are all the numbers that divide evenly into a number, including 1 butexcluding the number itself.

Example

1, 2, 3,4, 6, and, 12 are divisors of the number 12. They divide evenly into 12 withouta reminder.

But…the PROPERdivisors of 12 are: 1, 2, 3, 4, and 6; we should NOT include 12 here.

Let’s goback to “Amicable Numbers”.           

220 and284 are the smallest pair of amicable numbers.

The properdivisors of 220 are:

1, 2, 4,5, 10, 11, 20, 22, 44, 55, and 110

The sum ofall 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 nowfind the proper divisors of 284.

1, 2, 4,71, and 142

Add themup.

1 + 2 + 4+ 71 + 142 = 220

Excellent!Indeed they are friendly numbers!

Easiersaid than done… It’s a lot of work to find divisors of big numbers.

Here are afew tips to help you finding the divisors and sum of the divisors of a number.

# 1

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

Let’s lookat an easy example.

You canuse a factor tree to find the prime factors.

The primefactors of 24 are: 2, 2, 2, and 3

Let’scombine the factors in all possible ways to find the divisors of 24.

There arethree 2’s and one 3 in 24.

Let’sfirst combine the 2’s.

2 isdivisor of 24

2 x 2 = 4is a divisor of 24

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

Let’s nowcombine the 2’s and the 3.

3 is adivisor of 24

2 x 3 = 6is a divisor of 24

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

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

So, thedivisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

We include1 among the divisors, because ‘1 is a divisor of every number’.

Leavingout the 24 we get the PROPER divisors of 24.

The nextstep is to find the sum of proper factors of 24 which is:

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

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

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

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

# 2

Let ‘n’ bea positive integer.

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

Then thesum 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 tryusing this formula with the SAME SIMPLE example (24).

The primefactors of 24 are: 2, 2, 2, and 3

The primefactorization of 24 is (2^3) x (3^1).

So, thesum 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

REMEMBER60 is the sum of ALL the divisors of 24. But what we need is the sum of the PROPERdivisors of 24. So, SUBTRACT 24 from 60, we get:

60 – 24 =36

Hooray! Itworks! It matches with our answer!

Hope thishelps!