Blog Post #142: Numbers which complement each other as a friend

Thanks to the recent post by Dilip and my friend Sukumar for tagging me on that, which acted as the trigger for this next blog of mine. The numbers in limelight are 220 and 284 are part of what Mathematicians call as “Amicable Numbers” and these two are the smallest of that series

https://en.m.wikipedia.org/wiki/Amicable_numbers

Given the friendly nature of these amicable numbers, I was wondering if there are more attributes or qualities that these two numbers stand for and started exploring further this weekend. Here are few of my observations and findings

1. Add 220 and 284 and the resultant is 504 which is the smallest three digit number with factors from 1 to 9 except 5 ofcourse. When multiplied by 5 the resulting number is 2520 which is a special number - the lowest whole number which has factors from 1 to 10!

2. Subtract 220 from 284 and we get a perfect square

3. Both 220 and 284 are resultant of a perfect square minus 5. 220 is 225 - 5 while 284 is 289 - 5

4. Swap the last digit of these two numbers to form 224 and 280 which are successive multiples of 56! Truly amicable and friendly even when they swap a digit!

5. Let’s reverse both numbers and we get 022 and 482. Look at these closely and we realise that both are multiples of a prime number with 2! 11 x 2 and 241 x 2 !!

6. Suffix 9 to 220 and we get 2209 which is a perfect square of 47. So what’s big in that one may wonder…Replace the last digit 4 in 284 with 09 and we get 2809 which is a perfect square of 53. Guess what 53 and 47 in turn adds up to 100!!!

7. I am always fascinated by 1729 (Ramanujan number) and strongly believe 220 and 284 should have some relation with 1729. Guess what… 220 + 284 is 504 as we saw earlier and 504 + a perfect square (1225) yields 1729. Voila! That’s the brilliance of Ramanujan number

No wonder 220 and 284 are part of the Amicable or Friendly number list given they share so many common attributes as seen above. Draw a parallel to real life and/or corporate world, these are the qualities that we would look up for amongst friends or colleagues when they work together as a team - flexibility, fungibility, work towards a common goal, adapt to any ambiguous situation and produce results.

Coincidentally this is my 142nd blog which happens to be half of 284 :)