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The Arithmetic Magic Numbers

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:

2α = β
2β+1 = γ
2γ−100 = α

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

2α+1 = β
2β−100 = γ
2γ = α

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.

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.

Conclusion

The numbers having the property that multiplication by an integer gives a
shifting rearrangement of its digits are relative rarities.