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Number Play - The Magic Number of Kaprekar

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Magic Number of Kaprekar, or Kaprekar's Constant, is the number 61746174.

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To perform Kaprekar's routine, choose any four-digit number where at least two digits are different (e.g., 11211121 is allowed, but 11111111 is not).

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Arrange the digits of the number in descending order to create the largest possible number and in ascending order to create the smallest possible number.

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Subtract the smallest number from the largest number to get a new number.

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Repeat the process with the new number. Within a maximum of 77 steps, the operation will always result in 61746174.

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Once 61746174 is reached, the process will always result in 61746174 again: 7641−1467=61747641 - 1467 = 6174.

📐Formulae

Nlarge−Nsmall=NnextN_{large} - N_{small} = N_{next}

7641−1467=61747641 - 1467 = 6174

💡Examples

Problem 1:

Show the Kaprekar routine for the four-digit number 20242024.

Solution:

Step 1: Digits are 2,0,2,42, 0, 2, 4. Largest: 42204220, Smallest: 02240224. 4220−02243996\begin{array}{r} 4220 \\ - 0224 \\ \hline 3996 \end{array}

Step 2: Digits are 3,9,9,63, 9, 9, 6. Largest: 99639963, Smallest: 36993699. 9963−36996264\begin{array}{r} 9963 \\ - 3699 \\ \hline 6264 \end{array}

Step 3: Digits are 6,2,6,46, 2, 6, 4. Largest: 66426642, Smallest: 24662466. 6642−24664176\begin{array}{r} 6642 \\ - 2466 \\ \hline 4176 \end{array}

Step 4: Digits are 4,1,7,64, 1, 7, 6. Largest: 76417641, Smallest: 14671467. 7641−14676174\begin{array}{r} 7641 \\ - 1467 \\ \hline 6174 \end{array}

Explanation:

Starting with 20242024, we arrange the digits to find the difference. After 44 iterations, we reach the Kaprekar Constant 61746174.

Problem 2:

Why can we not use the number 55555555 for Kaprekar's routine?

Solution:

If we take 55555555, the largest number is 55555555 and the smallest number is 55555555. 5555−5555=05555 - 5555 = 0 Since the difference is 00, the routine cannot continue to reach 61746174.

Explanation:

Kaprekar's routine requires at least two distinct digits to ensure that the difference is not zero.