What is a defining property of a number that is divisible by 9?

A number is considered divisible by 9 if the sum of its digits is also divisible by 9. This property stems from the way numbers are structured in the decimal system. Each place value in a number represents a power of ten, and since ten is congruent to 1 modulo 9, the number itself is congruent to the sum of its digits modulo 9. Therefore, if the sum of the digits yields a total that is divisible by 9, the original number must also be divisible by 9.

For instance, consider the number 567. The sum of its digits is 5 + 6 + 7 = 18, and because 18 is divisible by 9, the number 567 is also divisible by 9. This rule simplifies checking for divisibility significantly and is a handy trick in arithmetic.

In contrast, simply having an even sum of digits, a sum that is a prime number, or a last digit of 9 does not guarantee the entire number's divisibility by 9. The defining property specifically relies on the sum of the digits being divisible by 9.