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Concerning Weighted Digit Sums and Remainders |
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Let the digital representation of a number n in base K be denoted as n_{K}. Suppose n_{K} is a two digit number aK+b. Its weighted digit sum with respect to a digit m is
Thus
Now suppose n=sm+r, where r is the remainder for the division of n by m and s is a positive integer. Then
That is, WDS_{m}(n_{K}) is a smaller multiple of m than n and has the same remainder upon division by m. If the process is continued until the result is less than K the remainder of the division of that result is the same as for the division n by m.
The remainder for the division of a number n by a digit m is the same as the division of the weighted digit sum of that number represented to base K larger than m.
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