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There is a number, 142857, with remarkable properties that are sometimes referred to as magic. These remarkable properties pertain to the shifting rearrangement of the digits in its multiples; i.e.,
The question is whether 142857 uniquely has these properties or whether there are other such numbers. Consider first the multiple by 2. Let α, β and γ be two-digit integers. Then the decimal number αβγ represents α(10,000)+β(100)+γ.
For 2(αβγ) to be equal to βγα requires:
This is a set of three linear equations in three unknowns. The constants are not all zero so there is a unique solution. It happens that the solution is known; i.e., α=14, β=28 and γ=57.
However, it is known that 2(285714)=571428. This corresponds to the solution of
This is the only rearrangement of the constant terms. So the magical shifting of the terms upon multiplication by 2 is only possible for 142857 and 285714. The number 142857 arises as a repeating block in the decimal representation of 1/7.
Consider the multiples of 076923.
The shifting rearrangement of the digits does not occur for the multiple of 2 but it does for the multiples of 3 and 9 From the previous analysis it is clear that 076923 and 230769 are the only 6-digit numbers for which the shifting rearrangement occurs for a multiplication by 3. The number 076923 arises as a repeating block in the decimal representation of 1/13.
The above statements refer to six-digit numbers. For four-digit numbers there may be other magic numbers.
The numbers having the property that multiplication by an integer gives a shifting rearrangement of its digits are relative rarities.
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