The number 142857 is a well-known mathematical curiosity. Here's a breakdown of its properties:
1. Cyclic Number
It is a cyclic number. When multiplied by 1 through 6, it produces the numbers 142857, 1428571, 14285712, ..., up to 142857123456. This property is due to its connection with the decimal expansion of 1/7, which is 0.\overline{142857}.
2. Multiplication Patterns
- 1 × 142857 = 142857
- 2 × 142857 = 285714
- 3 × 142857 = 428571
- 4 × 142857 = 571428
- 5 × 142857 = 714285
- 6 × 142857 = 857142
These results are cyclic permutations of the original number, which is why it's called a cyclic number.
3. Divisibility
- 142857 is divisible by 7 (since 1/7 = 0.\overline{142857}).
- It is also divisible by 3, 4, 6, 7, 12, 14, 21, 28, 42, 56, 84, and 168.
4. Special Properties
- 142857 × 9 = 1285713, which is a repeated cycle of the original number.
- It is not a prime number, but it is a highly composite number.
5. Applications
- It is used in cryptography and mathematical puzzles.
- It appears in recreational mathematics and number theory.
6. Interesting Fact
- If you write 142857 and multiply it by any number from 1 to 6, the result is a permutation of the original number.
Summary
142857 is a fascinating number due to its cyclic properties and its connection to the repeating decimal of 1/7. It is a cyclic number and has a rich history in mathematics.