Divisibility Rule: Does Being Divisible by 3 Imply Divisibility by 9?

Will a number divisible by 3 necessarily be divisible by 9?

To solve this problem, we need to understand the divisibility rules for 3 and 9:

• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 9 if the sum of its digits is divisible by 9.

Let's evaluate whether a number divisible by 3 is necessarily divisible by 9:

Consider the number 12. The sum of its digits is $1 + 2 = 3$, which is divisible by 3, so 12 is divisible by 3. However, when we check divisibility by 9, 12 is not divisible by 9 because 3 is not divisible by 9.

Now consider another number, like 18. The sum of its digits is $1 + 8 = 9$, which is divisible by both 3 and 9. Thus, 18 is divisible by both.

These examples demonstrate that while some numbers divisible by 3 are also divisible by 9 (e.g., 18), not all are (e.g., 12).

Therefore, a number being divisible by 3 does not necessarily mean it is divisible by 9.

The correct answer is No.