divisibleby9

Divisibleby9 refers to integers that are divisible by 9; equivalently, numbers n for which n ≡ 0 (mod 9). In decimal notation, this is commonly tested by the sum-of-digits rule: a base-10 number is divisible by 9 if and only if the sum of its decimal digits is divisible by 9. This stems from 10 ≡ 1 (mod 9), so a number n = d0 + 10 d1 + 10^2 d2 + ... satisfies n ≡ d0 + d1 + d2 + ... (mod 9). Repeating the digit-sum operation yields the digital root, which is 9 for nonzero multiples of 9 and 0 for 0.

Examples illustrate the rule: 45 has digits summing to 9, so it is divisible by 9; 12345 sums to 15, which is not a multiple of 9, so it is not divisible by 9; 9999 sums to 36, divisible by 9, so it is divisible by 9.

General notes and extensions: The digit-sum test is specific to base 10. In base b, a number is divisible by b−1 when the sum of its base-b digits is divisible by b−1, since b ≡ 1 (mod b−1). The divisibility by 9 can also be understood through modular arithmetic, including the fact that 9 divides 10^k − 1 for all positive integers k. Every multiple of 9 is also a multiple of 3, reflecting the relationship between divisibility by 9 and divisibility by 3.