A number divisible by 9 is always divisible by 3.
Understanding Divisibility by 9
The divisibility rule for 9 states that if the sum of a number's digits is divisible by 9, then the number itself is divisible by 9. This is because 9 is a multiple of 3 (9 = 3 x 3). Therefore, any number divisible by 9 will also have its digits sum to a multiple of 9, which inherently implies divisibility by 3.
Examples
- 18: The sum of the digits (1 + 8 = 9) is divisible by 9, and thus 18 is divisible by 9. Since 9 is divisible by 3, 18 is also divisible by 3 (18 / 3 = 6).
- 81: The sum of the digits (8 + 1 = 9) is divisible by 9, and therefore 81 is divisible by 9. 81 is also divisible by 3 (81 / 3 = 27).
- 279: The sum of the digits (2 + 7 + 9 = 18), which is divisible by 9, making 279 divisible by 9. 279 is also divisible by 3 (279 / 3 = 93).
Further Insights from References
Several references support this:
- One source explicitly states: "So, any number that is divisible by 9 is also divisible by 3 because 3 is a factor of 9."
- Other sources discuss the relationship between divisibility by 9 and rearrangement of digits, transposition errors, and the sum of digits, all indirectly confirming the inherent divisibility by 3. For instance, if a number is divisible by 9 (meaning the sum of its digits is a multiple of 9), then it must also be divisible by 3 (as 3 is a factor of 9).
