 Digit sum

The digit sum of a nonnegative integer in a specified radix is the sum of its numerical digits. Ifwrite $s_b(n)={\sum }_j{d}_j$, with $s_b(0)=0$. For example, $s_{10}(347)=3+4+7=14$. In binary representation, the digit sum is the Hamming weight of the digit string.

Since $b\equiv 1(\mathrm{mod}b-1)$, the positional expansion givesIn decimal representation, this yields the familiar divisibility tests by $3$ and $9$: the integer and its digit sum have the same remainder modulo either. For $347$, both $347$ and $14$ leave remainder $5$ modulo $9$. The argument explains the rule from modular arithmetic, rather than relying on a pattern noticed in examples.

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.