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The Interesting Properties of 1001,
Scheherazade's Number

When one hears the title 1001 Arabian Nights, one thinks of 1001 as being one unit beyond a very large number, 1000. And that may very well be the reason for the choice of the 1001, however there is another possible explanation. The number 1001 is the product of 7, 11 and 13, three consecutive prime numbers. One of these factors, 7, is consdered in some cultures to be a lucky. Likewise anther factor is 13, considered to be an unlucky number.

The simplicity of 1001 provides a method for determining whether a particular number N is a multiple of 7, 11 or 13. In this day of readily available high computing power such determination is trivial. However in the days when long division on paper was the only mode of computation this method would be of significance.

First note that q is a factor of N if and only if q is a factor of (N-kq) where k is any integer. So by subtracting multiples of q from N one can reduce the matter of factorization considerably. The long division is just such a process of subtracting multiples of a number from a test number. In the case of 1001 such long division is simple because the product of 1001 by a digit d is simply d00d.

Consider this example.


            8 413
       __________
1001  / 8,421,974 
        8,008
        _____
          4139
          4004
          ____
           1357
           1001
           ____
            3564
            3003
            ____
             561

The remainder 561 can be quickly tested to see if it is a multiple of 7, 11 and/or 13. It is not a multiple of 7. It is not a multiple of 13, but it is a multiple of 11.

Scheherezade!


And one can see that if the remainder had been 560 the N would be a multiple of 7. For the remainder to be 560 the number N would have to be one less than 8,421,974; i.e., 8,421,973. Likewise if the remainder had been 562 the number N would have been a multiple of 13. This would require that N be 8,421,974+1; i.e., 8,421,975.


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