Wednesday, May 28, 2014

Numbers and Their BFFs

Amicable numbers. First off, what are they? Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number (thank you Wiki). But what does that exactly mean? Let's look at an example:

To show that two numbers are amicable you must first find their proper divisors. So first you need to know what a proper divisor is: a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3 (again, thank you Wiki). Let's work with 220 and 284.
Proper divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
Proper divisors of 284: 1, 2, 4, 71, 142

If we add together all the proper divisors of 220 we obtain 284.
If we add together all the proper divisors of 284 we obtain 220.

Weeeeeird. In short, that is what amicable numbers, or more commonly amicable pairs, are.

This site gives a nice little brief history on amicable numbers. Briefly discuss famous mathematicians who found amicable pairs, Pythagoras being the first to find 220 and 284 as an amicable pair, and several other devoting their time to finding others. The page even offers a way to discover other amicable pairs. Fancy.

Who knew that numbers had BFFs. I guess everyone deserves them.

  • There are some amicable pairs (m, n), in which the sum of digits of m and n is equal [4]. For example, consider amicable pair ( 69615, 87633), 
    • Sum of digits of 69615 = 6+9+6+1+5 = 27 
    • Sum of digits of 87633 = 8+7+6+3+3 = 27 
  • There is no amicable pair in which one of the two numbers is a square

1 comment:

  1. Complete: Be sure to list your ref.s. (wiki?)
    Content: so how rare or common are these? Do you know why there's no amicable numbers where one is a square?
    Consolidated: be a good one for so what or now what.
    clear, coherent:+