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 Description  
 Amicable Numbers:  Two numbers are amicable if each is the sum of the other number's proper divisors.
Amicable in everyday language means "friendly," so amicable numbers are also considered "friendly" numbers.

Examples of Amicable Numbers
Susan, who has been exploring with amicable numbers, tells us that the numbers 220 and 284 are amicable. But we're not so sure. To find out if Susan is right, let's use a factor tree and find each number's prime factorization.

For 220,



Combining all the possibilities, we get the factors for 220 are: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110,and 220. The sum of these factors, except for 220, equals 284.

For 284,



Combining all the possibilities, we get the factors for 284 are:

1, 2, 4, 71, 142, and 284. The sum of these factors, except for 284, equals 220.

So, 220 and 284 are very amicable indeed!

Non-Example of Amicable Numbers
Are 12 and 16 amicable numbers? The number 12 has the positive factors 1, 2, 3, 4, 6, and 12. The sum of the proper factors (all positive factors not including 12) is 1+2+3+4+6 = 16. The number 16 has the positive factors 1, 2, 4, 8, and 16. The sum of the proper factors (all positive factors not including 16) is 1+2+4+8 = 15. Not quite amicable!

Interesting to Note:

The pair of numbers 220 and 284 was once thought to be magical. As a result, the pair played an important role in magic, sorcery, astrology, and horoscopes. For example, according to superstition, two talismans (ornaments or charms) on which these numbers were engraved would guarantee a perfect friendship between the people who wore them.