Explanation: In number theory, divisibility rules are fundamental principles that help determine whether a number is divisible by another number without performing the actual division. One such rule is the divisibility rule for 9, which states that a number is divisible by 9 if the sum of its digits is divisible by 9. For example, the number 189 is divisible by 9 because the sum of its digits (1 + 8 + 9 = 18) is divisible by 9.
The relationship between divisibility by 9 and divisibility by 3 is based on the fact that 9 is a multiple of 3. Specifically, 9 can be expressed as 3 times 3 (9 = 3 × 3). Therefore, if a number is divisible by 9, it can be divided by 9 without leaving a remainder. Since 9 is a multiple of 3, the number can also be divided by 3 without leaving a remainder. This is why a number divisible by 9 is also divisible by 3.
To illustrate this concept, consider the number 189. We have already established that 189 is divisible by 9. To check divisibility by 3, we can sum the digits of 189 (1 + 8 + 9 = 18), and since 18 is divisible by 3, 189 is also divisible by 3. This demonstrates the inherent relationship between divisibility by 9 and divisibility by 3.
It is important to note that while divisibility by 9 implies divisibility by 3, the reverse is not necessarily true. A number divisible by 3 is not always divisible by 9. For example, the number 12 is divisible by 3 but not by 9. This distinction is crucial in understanding the specific relationships between different divisibility rules.
In summary, the correct answer is that a number divisible by 9 is also divisible by 3. This is because 9 is a multiple of 3, and any number divisible by a multiple of 3 is also divisible by 3 itself. Understanding these relationships is essential for solving problems involving divisibility and number theory.