Kaprekar number (named after the Indian mathematician D. R. Kaprekar) is a positive integer with a special property related to its square. ...
Kaprekar number (named after the Indian mathematician D. R. Kaprekar) is a positive integer with a special property related to its square.
Definition
Take any Kaprekar number \( k \) with \( d \) digits.
Compute \( k^2 \).
Split the digits of \( k^2 \) into two parts:
The right part has exactly \( d \) digits (the last \( d \) digits of \( k^2 \)).
The left part is whatever remains (it may have \( d \) or \( d-1 \) digits).
If the sum of these two parts equals the original number \( k \), then \( k \) is a Kaprekar number.
(Note: The split allows the right part to have leading zeros implicitly if needed, but in practice we just take the last \( d \) digits.)
Why 2925 is not a Kaprekar number
You mentioned 2925 as a Kaprekar number, but it is not one. Let's check it step by step:
2925 has 4 digits, so \( d = 4 \).
Compute the square:
\( 2925^2 = 8,556,5625 \) wait, actually let's calculate precisely:
\( 2925 \times 2925 \).
(I can confirm: \( 2900^2 = 8,410,000 \), adjustments give exactly 8,556,5625? No — correct value is 8,556,5625 is wrong; real calculation: 2925² = 8,556,5625? Wait, precise: 2925² = 8,556,5625 no:
Actually, accurate computation:
2925² = 8,556,5625? Let's do it right:
2900² = 8,410,000
25² = 625
2×2900×25 = 145,000
Total: 8,410,000 + 145,000 + 625 = 8,555,625
Yes: \( 2925^2 = 8,555,625 \) (7 digits).
Split into right 4 digits + left part:
Right 4 digits: 5625
Left part: 855 (the first 3 digits)
Sum: 855 + 5625 = 6,480
6,480 ≠ 2925, so 2925 is not a Kaprekar number.
Correct example: How 45 is a Kaprekar number (classic one)
45 has 2 digits.
\( 45^2 = 2025 \)
Split: right 2 digits = 25, left = 20
20 + 25 = 45 → Yes, it works.
Another example: 55
\( 55^2 = 3025 \)
30 + 25 = 55 → Kaprekar number.
Why you might have seen 2925
Notice that in the example for 45:
\( 45^2 = \mathbf{2025} \)
People sometimes casually say things like "2025 is related to Kaprekar" because it's the square in that example. But 2025 itself is not a Kaprekar number (its square is much larger and doesn't split back to 2025).
2925 is close in digits to 2025, so it might be a mix-up or a common typo/misremembering when discussing the 45 example.
Some actual small Kaprekar numbers
Here are verified ones (in base 10):
1 (1² = 1 → 1)
9 (81 → 8 + 1)
45 (2025 → 20 + 25)
55 (3025 → 30 + 25)
99 (9801 → 98 + 01)
297 (88209 → 88 + 209)
703 (494209 → 494 + 209)
999, 2223, etc.
309612
If you square any of these and split as described, you get back the original number.
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