What are Amicable Numbers? (Communication Math)

When I first saw this question I really didn't know what it meant. This is not something that I have heard of, or at least remembered about. Therefore, I had to do some digging and figure out what this means. I found a couple different definitions for this.

Amicable numbers are a pair of numbers such that the sum of their proper divisors (not including itself) equals the other number.

After reading this definition I thought I knew what they were talking about, but as I started to try and come up with some numbers I started having a hard time. Therefore, I had to research some examples before I could start to come up with my own!

(220, 284)- These numbers are said to be amicable numbers.

First we will take 220 and come up with all the proper divisors

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

If we add these numbers together 1+2+4+5+10+11+20+22+44+55+110=284

Now, we can come up with all the proper divisors for 284

1, 2, 4, 71, 142

If we add these numbers together 1+2+4+71+142=220

Therefore, these numbers are amicable numbers, or amicable pairs.

After looking at this example it really seemed to help. But, I couldn't help but wonder how difficult this would be to just pick two numbers. It would take me forever to pick two numbers and see if they were an amicable pair. So, I decided that I would look for another pair and see if there was a pattern.

We will take 2620 and come up with the proper divisors.

1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310

If we add these together 1+2+4+5+10+20+131+262+524+655+1310=2924

Now, we take 2924 and come up with the divisors

1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462

We add these together 1+2+4+17+34+43+68+86+172+731+1462=2620

Therefore, (2620, 2924) are an amicable pair.

I figured out that even after having the numbers of the amicable pair, it was difficult to find all the divisors. That actually took me a long time to do. I had to think about all the rules you learned in high school dealing with the different test.

This was a lot of fun to look up and learn. I had never heard of this before so it was interesting to learn!

complete: to make an exemplar, I'd add a bit more. Either about amicable numbers from your reading, or add your thinking and methods to the 2nd example. An example you tried of a non-amicable pair would be good, too. Sounds a little like you just picked and got lucky.

ReplyDeleteO/w: good post on a quirky topic.