So far we have discussed Kaprekar's Constant for two,three and four digit numbers. Is it all over?? No....not yet...in fact, we will certainly obtain such Kaprekar's Constants or Series as we proceed further. But, we do not find unique Kaprekar's Constant anymore. Scientists are still working on this fact...the search for the third unique Kaprekar's constant is still on...the first two being 495 and 6174.
Here are a few examples of how Kaprekar's Constant for more digits look like:
Digits ---------- Kaprekar's Constant(s)
5 -------------- None
6 -------------- 549945, 631764
7 -------------- None
8 -------------- 63317664, 97508421
9 -------------- 554999445, 864197532
10 ------------- 6333176664, 9753086421, 9975084201
[Source: Mysterious number 6174]
One interesting thing that we can observe here:
Kaprekar's Constant for four-digit numbers=6174
One of the Kaprekar's Constant for six-digit numbers=631764
One of the Kaprekar's Constant for eight-digit numbers=63317664
One of the Kaprekar's Constant for ten-digit numbers=6333176664.....and so on.
This observation doesn't have any consequences in our search...I just sited it as an interesting fact.
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The "one more interesting thing" was really interesting. I just found another interesting thing:
ReplyDeletethe sum of digits of a kaprekar's constants is always a multiple of 9. In fact it turns out to be 45 for ALL three Kaprekar's constant for 10-digit nos.
see...whenever u substract cba from abc, or dcba fro abcd...the resulting no. is always a multiple of 9. so it's sum of digits must be divisible by 9....but I haven't noticed the 45 fact...it's interesting...thanks! I will try to work on it...
ReplyDeleteyeah i later realized that that thing was obvious.
ReplyDelete