San José State University

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Thayer Watkins
Silicon Valley
& Tornado Alley
USA

Magic Numbers

The number 142857, which is involved in the decimal representation of 1/7, has some interesting properties, enough such it is sometimes called a magic number. Consider its multiples.

The Multiples of 142857
MultiplierMultipleComment
1142857 
2285714First two digits moved to end
3428571First digit moved to the end
4571428First four digits moved to the end
5714285First five digits moved to the end
6857142First three digits moved to the end
7999999 
81142856One deducted from the last digit
and moved to the front
9128571313 deducted from the initial 14
and moved to the end

Another magic number is 076923, which comes from the decimal representation of 1/13.

The Multiples of 076923
MultiplierMultipleComment
1076923 
2153846 
3230769First four digits moved to the end
4307692Last digit moved to the front
5384615 
6461538 
7538461 
8615384 
9692307First two digits moved to the end
10769230First digit 0 moved to the end
11846153 
12923076First three digits moved to the end
13999999 


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