Amicable numbers

Two numbers are said to be amicable if each number is the sum of the proper divisors of the other.

A proper divisor of a number is any divisor of the number except the number itself.

For example, the proper divisor of 12 are 1, 2, 3, 4, and 6.

Notice that 12 is not included in the list although it is a divisor of 12.

Now we will show that 220 and 284 are amicable numbers.

The proper divisors of 284 are 1, 2, 4, 71, and 142

1 + 2 + 4 + 71 + 142 = 220

The sum of the proper divisors of 284 is equal to 220

The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

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

The sum of the proper divisors of 220 is equal to 284.

Therefore, we can conclude that 220 and 284 are amicable numbers.

A short list of amicable numbers are shown below

220                     284

1184                  1210

2620                  2924

5020                  5564

6232                  6368

10744               10856

12285              14595


......

.......


437456              455344

469028              486178

503056              514736

522405              525915

600392              669688

609928             686072

Let us show that 600392 and 669688 are amicable

Use this calculator to quickly see the factors. That way you don't have to waste time.



The proper factors of 609928 are   1,  2,  4,  8,  11,  22,  29,  44,  58,  88,  116,  232,  239,  319,  478,  638,  956,  1276,  1912,  2552,  2629,  5258,  6931,  10516,  13862,  21032,  27724,  55448,  76241,  152482,  and 304964

Add all the numbers and it will equal to 686072

The proper factors of 686072 are 1,  2,  4,  8,  191,  382,  449,  764,  898,  1528,  1796,  3592, 85759,  171518,  and  343036

Add all the numbers above and you will get 600392

600392 and 669688 are indeed amicable





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