The \( 6174 \) Phenomenon

For any \( 4 \)-digit number where not all the places have the same number, repeat the calculation of subtracting the smallest number from the largest number that can be obtained by rearranging the numbers in each place, and you will always arrive at \( 6174 \) at the end.

This amazing fact was discovered by the Indian mathematician Kaprekar, and the number \( 6174 \) is named after him as the Kaprekar number.

Example Calculation

Let’s try this with \( 1234 \): \[ 4321 – 1234 = 3087 \] \[ 8730 – 0378 = 8352 \]\[ 8532 – 2358 = 6174\ ! \]

Kaprekar Number and Multiples of \( 9 \)

Additionally, it can be observed that the Kaprekar number is a multiple of \( 9 \). In fact, in general, for a \( 4 \)-digit number \( bcad \) \( (a \ge b \ge c \ge d) \)

\[ (\text{Maximum number}) = 1000a + 100b + 10c + d \] \[ (\text{Minimum number}) = 1000d + 100c + 10b + a, \]

So, \[ (\text{Maximum number})-(\text{Minimum number}) = 999a + 90b-90c-999d. \]

This result is clearly a multiple of \( 9 \).