mod9

Mod9, also known as the modulo operation by 9, is a mathematical operation that finds the remainder after division by 9. For any integer n, the result of n mod 9 is an integer r such that n = 9q + r, where q is an integer and 0 <= r < 9. This operation is frequently used in computer science and number theory.

One notable property of modulo 9 is its connection to the sum of the digits of a number. For any positive integer, the number modulo 9 is equal to the sum of its digits modulo 9. For example, consider the number 123. The sum of its digits is 1 + 2 + 3 = 6. 123 divided by 9 gives a remainder of 6. Similarly, for 457, the sum of digits is 4 + 5 + 7 = 16. 16 mod 9 is 7, and 457 mod 9 is also 7. This property can be used as a quick divisibility test for 9: if the sum of a number's digits is divisible by 9, then the number itself is divisible by 9.

The modulo 9 operation is a specific instance of the more general modulo operation (mod n). It has applications in checksum algorithms, hashing functions, and in simplifying calculations where only the remainder after division by 9 is of interest. The behavior of modulo 9 is well-defined for both positive and negative integers, though the specific representation of the remainder for negative numbers can vary between programming languages.