Amicable Numbers
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!
