The digit sum of a nonnegative integer in a specified radix is the sum of its numerical digits. Ifwrite , with . For example, . In binary representation, the digit sum is the Hamming weight of the digit string.
Since , the positional expansion givesIn decimal representation, this yields the familiar divisibility tests by and : the integer and its digit sum have the same remainder modulo either. For , both and leave remainder modulo . The argument explains the rule from modular arithmetic, rather than relying on a pattern noticed in examples.
The alternating digit sum of a nonnegative integer in radix iswith a positive sign on the units digit. Starting the signs from the opposite end can change the overall sign, so the convention should be stated.
Because , the positional expansion gives . This proves a divisibility test: divides the number exactly when it divides its alternating digit sum. In decimal notation, has alternating sum and equals .
The rule differs from the ordinary digit sum test modulo , which instead uses . Both follow from the same modular arithmetic principle of replacing the radix by its residue before evaluating the digit polynomial.
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The digit sum, also known as the sum of digits, is calculated by adding together all the individual digits of a given number. For instance, if you take the number 1234, the digit sum would be calculated as follows: 1 + 2 + 3 + 4 = 10 In this case, the digit sum of 1234 is 10. The concept is often used in various areas of mathematics, including number theory and numerical analysis.