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
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