Blog Post #125: Unique attributes of the number 125 - The nonchalant allrounder!

This is my 125th blog post and decided to pen down about the number 125 itself as it has some very unique attributes apart from some of the very obvious ones

Drawing a parallel to the corporate world, 125 is like the all-rounder in any project team - multi-skilled, possess unique attributes compared to its peers and more importantly don different avatars or wear various hats when confronted with different external situations and unknowns

1. For starters, 125 is 5 ^ 3 and let's table it as the base attribute to begin with

2. 125 can be written as the sum of multiple squares (combination of various skillsets) and few are listed below

     a) 100 + 25 = 125

     b) 121 + 4  = 125

3. 125 has a very rare attribute which not many numbers can boast of

     125 is 5 ^ 3, Sum of the digits 1 + 2 + 5 is 2 ^ 3 and 512 which is an anagram of is 8 ^ 3 (can be also written as (1 + 2 + 5) ^ 3

4. Further more look at the following pattern sequence where you prefix 1 to 9 to 125 - drawing a parallel to how an individual would react to varying situations while handling a project milestone

    a) 1125 = 35 ^ 2 - 10 ^ 2

    b) 2125 = 45 ^ 2 + 10 ^ 2

    c) 3125 = 55 ^ 2

    d) 4125 = 65 ^ 2 - 10 ^ 2

    e) 5125 = 75 ^ 2 - 20 ^ 2 - 10 ^ 2

    f) 6125 =  75 ^ 2 + 20 ^ 2 + 10 ^ 2

    g) 7125 =  85 ^ 2 - 10 ^ 2

    h) 8125 =  85 ^ 2 + 30 ^ 2

    i) 9125  =  95 ^ 2 + 10 ^ 2

  5. And finally the blockbuster - 125 squared = 15625. Let's go deeper one layer after another 

      a) 15625 = 125 ^ 125

      b) Peel off 1 and you get 5625 which is 75 ^ 2

      c) Peel of 5 and resultant is 625 which is 25 ^ 2

      d) Peel of 6 and resultant is 25 which is 5 ^ 2

It is not just the breadth of skills that someone exhibits but also the depth of understanding, which is important as we peel/remove one layer after another in the scenario above, that makes someone a true all-rounder and Man Friday that any project team would love to have!

#MatheMagic #AnandMathsBlog 

 

Blog Post #124: Revisiting good old Armstrong numbers

Few days back, I was interacting with about 10-15 students last week of a Government school over video call and one of the topic was around 3 digit Armstrong numbers and what makes them special. It made a fascinating discussion with the kids as they started looking for out-of-the-box answers

That inspired me to revisit these numbers and look at some interesting patterns that emerge

Let's look at all the 3 digit Armstrong numbers to begin with - 153, 370, 371, 407

For starters, these are called so, as the sum of their digits raised to the power 3 equals to the number itself

1^3 + 5^3 + 3^3 for e.g is 153. 

For this post let's consider 153 and its anagrams and we would be amazed by the unique characteristics they demonstrate

a) 153 also happens to be sum of two squares (12 ^ 2 + 3 ^ 2) and difference of two squares (13 ^2 - 4 ^ 2)

b) 135 which is an anagram of 153 is a difference of 12 ^ 2 - 3 ^ 2

c) 351 which is also an anagram of 153 is a sum of two cubes - 7 ^ 3 + 2 ^ 3

d) 513, another anagram of 153 is sum of two cubes - 8 ^ 3 + 1 ^ 3 and difference of two squares - 23 ^ 2 - 4 ^ 2

e) 315 is difference of two squares again 18 ^ 2 - 3 ^ 2!

f) Now let's add two perfect squares to 153 and see what happens with the resulting numbers

153 + 81 = 234 and 153-144 = 09 and when we concatenate 234 and 09 we get 23409 which in turn is 153 * 153! (153 squared)

In summary, 153 is a truly multi-faceted number and can don various hats with ease, adapting to external conditions