Understanding Divisibility Rules: Numbers Divisible by 3 but Not by 9 and Vice Versa

Divisibility rules are useful mathematical tools that help us quickly determine whether a number is divisible by another number without performing division. Specifically, we will explore the conditions for a number to be divisible by 3 but not by 9, and vice versa. We will also discuss the underlying principles and provide examples to clarify these concepts.

A Quick Overview of Divisibility Rules

There are two key divisibility rules related to 3 and 9.

A number is divisible by 3 if the sum of its digits is divisible by 3. For example, the number 1234 (1 2 3 410, 10 is not divisible by 3) is not divisible by 3, but 1236 (1 2 3 612, 12 is divisible by 3) is.

A number is divisible by 9 if the sum of its digits is divisible by 9. For example, 27 (2 79, 9 is divisible by 9) is divisible by 9, but 26 (2 68, 8 is not divisible by 9) is not.

Numbers Divisible by 3 but Not by 9

Any number that satisfies the divisibility rule for 3 but does not satisfy the rule for 9 falls into this category.

For a number to be divisible by 3, it must have a sum of digits divisible by 3. For it to not be divisible by 9, the sum of its digits must not be divisible by 9.

In this context, we can say:

6, 12, 15, 21, 24, 30, 33, 36, 42, 45, 48, 51, 54, etc. These numbers are divisible by 3 because the sum of their digits is divisible by 3, but they are not divisible by 9 because the sum of their digits is not divisible by 9. For example, 36 (3 69) is divisible by 3, but not by 9.

Numbers Divisible by 9 but Not by 3

It's important to point out that there is a fundamental principle here: 9 is a multiple of 3. This means that any number divisible by 9 is also divisible by 3. Therefore, there are no numbers that are divisible by 9 but not by 3.

For example, take the number 18 (1 89, 9 is divisible by 9), 18 is divisible by both 3 and 9. Another example is 27 (2 79, 9 is divisible by 9); similarly, 27 is divisible by both 3 and 9.

Underlying Principles and Numbers

The relationship between 3 and 9 is essential here. Since 9 32, every multiple of 9 is also a multiple of 3. Therefore, no number that is divisible by 9 can be divisible by 3 without being divisible by 9.

Illustrative Examples

To further illustrate, we can use the following examples:

Example 1: Consider 1236. The sum of its digits is 12 (1 2 3 612), which is divisible by 3 but not by 9. Therefore, 1236 is divisible by 3 but not by 9.

Example 2: Let's take the number 6. Since 6 (6) is divisible by 3 (as 6 is divisible by 3), and also (6 is not divisible by 9 as 6 is not divisible by 9).

Mathematical Explanation

Let's delve into the mathematical explanation. We can say that if x is a number not divisible by 3, then 3x y. Here, y is a number divisible by 3 but not by 9.

For instance, if x 124578, then 3x 373734. Here, 373734 is divisible by 3 (since the sum of its digits is 30, which is divisible by 3) but not by 9 (as the sum of its digits is 30, which is not divisible by 9).

Conclusion

In summary, numbers divisible by 3 but not by 9 can be identified by their digits' sum being divisible by 3 but not by 9. Conversely, there are no numbers that are divisible by 9 but not by 3, due to the fundamental relationship between these numbers.

Understanding these rules can help in various mathematical computations and problem-solving scenarios. By mastering divisibility rules, one can quickly assess the divisibility of numbers, improving efficiency and accuracy in calculations.

Remember, the key is to keep the relationship between 3 and 9 in mind: any multiple of 9 is inherently divisible by 3.