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Amicable Partners

Date: 07/23/98 at 23:07:39
From: keelan
Subject: Amicable partners

1) Find the factors of 1184 and find its amicable partner.

2) Choose another number between 100 and 1000 and show that it does
not have an amicable partner.

Date: 07/25/98 at 01:46:37
From: Doctor Schwa
Subject: Re: Amicable partners

Amicable partners are pairs of numbers such that the factors of one,
excluding the number itself, sum to make the other. So if we can add
up the factors of 1184, then we know its amicable partner (though it
would be good to check that the relationship works both ways as it's
supposed to).

There are many ways to list the factors of a number. One way is to
start at both ends and work toward the middle, like:

1, 1184
2, 592
4, 296

and so on. Note that when we add them up, we would not include 1184 in
the sum.

Another way is to look at the prime factorization of a number, for
instance 120 = 2^3 * 3 * 5. Then its factors will be all of the ways
that we can combine the prime factorization, such as some twos, times
maybe a three, and maybe a five. In other words:

1, 2, 4, or 8
times 1 or 3
times 1 or 5

Multiplying through the list we get:

1 times 1 times 1 = 1
2 times 1 times 1 = 2
4 times 1 times 1 = 4
8 times 1 times 1 = 8
1 times 3 times 1 = 3
2 times 3 times 1 = 6
4 times 3 times 1 = 12
8 times 3 times 1 = 24
.
.
.
1 times 3 times 5 = 15
2 times 3 times 5 = 30
4 times 3 times 5 = 60
8 times 3 times 5 = 120 (Note: we do not include this in the sum)

This is a systematic way for us to find all of the factors. Using this
I get:

1, 2, 4, 8; 3, 6, 12, 24; 5, 10, 20, 40; 15, 30, 60, 120

I'll leave it to you to use one of these methods to find the factors
of 1184, excluding 1184 itself, add them up, and then check that the
other number's factors sum to 1184.

For question 2, we'll consider 120, which I just used. We list all of
the factors of 120 above. Now we need to add up all those factors,
excluding 120. I get:

(1+2+4+8) + 3*(1+2+4+8) + 5*(1+2+4+8) + 3*5*(1+2+4)
= 15 + 3*15 + 5*15 + 3*5*7
= 15 * (1 + 3 + 5 + 7)
= 15 * 16 = 240

This is a very convenient method for adding up factors. You might want
to try it for 1184. Now to show that 120 has no amicable partner, you
just need to show that the factors of 240, excluding 240, don't add up
to 120. But since 1 and 120 are both factors of 240, we're already too
big, and there's no way that the factors of 240 can add up to 120.

I hope that helps clear things up!

- Doctor Schwa, The Math Forum
Check out our web site! http://mathforum.org/dr.math/

Associated Topics:
High School Number Theory

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