Amicable Numbers

A pair of two different non-zero numbers are said to be Amicable numbers/pair, if the sum of the proper divisors of each number is equal to the other number in the pair.

A proper divisor of a number is a positive factor of that number except the number itself. For example:- Divisors/Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. But proper divisors of 24 are 1, 2, 3, 4, 6, 8 and 12 (24, i.e the number itself is not included).

Let’s observe the pair of two numbers 220 and 284. They are amicable numbers.

Proper divisors of 220 – 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

Sum of proper divisors of 220 – 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

Proper divisors of 284 – 1, 2, 4, 71, 142

Sum of proper divisors of 184 – 1 + 2 + 4 + 71 + 142 = 220

In the above example, the sum of the proper divisors of a number is equal to the other number. Hence, 220 and 284 are amicable numbers. 220 and 284 is the smallest or first pair of amicable numbers.

Note:- For a pair of two numbers P and Q,

P = sum of proper divisors of Q and

Q = sum of proper divisors of P

List of some amicable pairs

(220, 284) (1184, 1210) (2620, 2924) (5020, 5564) (6232, 6368) (10744, 10856) (12285, 14595) (17296, 18416) (63020, 76084) (66928, 66992)

Fascinating facts about amicable numbers

  1. Both the numbers in a pair of amicable numbers are either odd or either even.
  2. Basically, it is not known whether, there exists a pair of amicable numbers with one even and one odd number.
  3. No amicable pair exists, where one of the two numbers is a square.
  4. There are some amicable pairs, where the sum of the digits of both numbers is equal. For example – 100485 and 124155 (1 + 0 + 0 + 4 + 8 + 5 = 18 = 1 + 2 + 4 + 1 + 5 + 5). Some more pairs like this are, (69615, 87633) (1358595, 1486845) (131483835, 132692805)
  5. There are some amicable pairs, where both the numbers in the pair are completely divisible by the sum of their digits. For example – (2620, 2924). 2620 is completely divisible by the sum of its digits, i.e. 10 (2 + 6 + 2 + 0 = 10) in 292 times and 2924 is completely divisible by the sum of its digits, i.e. 17 (2 + 9 + 2 +4 = 17) in 172 times.

HAPPY LEARNING……………………..

Share this