Two numbers are amicable or “friendly” if each is equal to the sum of the natural number divisors of the other. This is excluding the numbers themselves.

The first pair of amicable numbers, (220 & 284) are said to have been discovered by Pythagoras. They were thought of as having magical powers!

The divisors of 284, excluding itself are:

1, 2, 4, 71 & 142

When we add these divisors together we have:

1 + 2 + 4 + 71 + 142 = 220

Now, the divisors of 220, excluding itself are:

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

When we add these divisors together we shave:

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

Amicable numbers have a long history in magic and astrology and were used for preparing horoscopes, making love potions and talismans.

There is even an ancient Jewish commentary that Jacob gave his brother 220 sheep, (200 females and 20 males) when he was afraid his brother was going to kill him!

Ever since the discovery of this first pair of amicable numbers, mathematicians have sought out and discovered more than 40,000 amicable pairs, 7,500 of which contain 10 or fewer digits:

  • Euclid discovered 59 amicable numbers, among them, the pair (6,232 & 6368) and also the pair, (10,744 & 10, 856).
  • Descartes found the following pair, (9,363,584 & 9,437,056).
  • Remarkably many years after Euclid, a 16-year-old Italian boy, Nicolo Paganini, found what is now known as the second smallest pair of amicable numbers, (1,184 & 1,210).

Here are the first six pairs of Amicable numbers:

220 & 284

1,184 & 1,210

2,620 & 2,924

5,020 & 5,564

6,232 & 6,368

10,744 & 10,856

A curious feature of Amicable numbers is that there none are square numbers.

Each of the following amicable pairs have an additional common property.

(69,615 & 87,633), (100,485 & 124,155), (1,358,595 & 1,486,845)

Can you identify this property?

Click to see solution..

Solution

In each case, the sum of their digits are equal

(69,615 & 87,633) sum of their digits = 27
(100,485 & 124,155) sum of their digits = 18
(1,358,595 & 1,486,845) sum of their digits = 36