Thursday, September 25, 2008

6174

6174 = 2 x 32 x 73.

6174 has four 4s in base 5: 144144.

6174
is Kaprekar’s Constant, named for D.R. Kaprekar of Devlali, India.

Choose a four-digit number, as long as the four digits are not all identical. Rearrange the four digits to get the largest and smallest numbers these digits can make. Subtract the smallest number from the largest to get a new number. Repeat these operations for each new number. For any four-digit number, the process produces 6174 in at most seven steps, and the calculation repeats.

5200 – 0025 = 5175
7551 – 1557 = 5994
9954 – 4599 = 5355
5553 – 3555 = 1998
9981 – 1899 = 8082
8820 – 0288 = 8532
8532 – 2358 = 6174
7641 – 1467 = 6174

6174 is also a Harshad (or Niven) number, which is divisible by the sum of its digits.

On the standard 33-hole cross-shaped peg solitaire board, 6174 is the number of distinct board positions after 9 jumps that can still be reduced to one peg at the center, starting with the center vacant.

The square root of the sums of the squares of the prime factors of 6174 is a prime: 22 + 32 + 32 + 72 + 72 + 72 = 169 = 132, and 13 is a prime.