Today's number is 142857
142857 is one of the most famous "curious numbers" in mathematics. It appears as the repeating digits of 1/7, and it has several surprising properties that make it worth remembering.
Cyclic Number
This number is the best known cyclic number in base 10. It is the six repeating digits of 1/7 (0.\overline{142857}) If you multiply 142857 by 2, 3, 4, 5, or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2/7, 3/7, 4/7, 5/7, and 6/7 respectively:
1 * 142,857 = 142,857 -> 1/7 = 0.\overline{142857}
2 * 142,857 = 285,714 -> 2/7 = 0.\overline{285714}
3 * 142,857 = 428,571 -> 3/7 = 0.\overline{428571}
4 * 142,857 = 571,428 -> 4/7 = 0.\overline{571428}
5 * 142,857 = 714,285 -> 5/7 = 0.\overline{714285}
6 * 142,857 = 857,142 -> 6/7 = 0.\overline{857142}
7 * 142,857 = 999,999 -> 7/7 = 0.\overline{999999} = 1
If multiplying by an integer greater than 7, there's a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits to the remaining digits and repeating this process until only 6 digits are left, it will result in a cyclic permutation of 142857. For example, 815 * 142,857 = 116428455
Take the last 6 digits (428,455), and add them to the remaining digits:
116 + 428,455 = 428,571, which is a cyclic permutation.
This happens because 10^6 is congruent to 1 (mod 142857). Because of that, replacing a block of digits by the sum of its six-digit chunks preserves the remainder modulo 142857.
Mental Division
I've actually used this trick before when I needed to divide by 7 quickly. If you have some natural number, it's fairly trivial to find the integer part. Once you have it, you can then divide the remainder by 7 and consider 142857. If your remainder is n, you look at n/7 in the table above.
For example, to compute 103 / 7, first note that 14 * 7 is 98, leaving a remainder of 5. Then note that 5 / 7 = 0.\overline{714285}.
So the 103 / 7 = 14.\overline{714285}.
Kaprekar Number
142857 also appears in a completely different corner of recreational mathematics.
A p-Kaprekar number is a number whose square can be split into two parts (whose second part is p-digits long) and add to the original number. For example, 45 is a 2-Kaprekar number because 45^2 = 2025, and 20 + 25 = 45.
142857 is a 6-Kaprekar number since 142857^2 = 20,408,122,449, and 20,408 + 122,449 = 142,857.
Conclusion
Overall, 142857 is a pretty cool number. So the next time you get mugged and are asked to find 7352 / 7 to at least six decimal points of accuracy, you won't be in trouble.
