Divisibility rules are simple techniques used to determine whether a given number is divisible by another number without actually performing the division. They are useful tools in mathematics and can help students quickly identify whether a number is divisible by another number. In this article, we will explore the divisibility rule for 12, its derivation, and some examples of how it can be used.

## What is Divisibility?

Divisibility refers to the ability of one number to divide another number without leaving a remainder. For example, 6 is divisible by 2 because 6 can be divided into 3 groups of 2 without leaving a remainder. Similarly, 12 is divisible by 3 because 12 can be divided into 4 groups of 3 without leaving a remainder.

## The Divisibility Rule for 12

To determine if a number is divisible by 12, we need to examine the last two digits of the number. If the number formed by the last two digits of the given number is divisible by 12, then the entire number is also divisible by 12.

## Derivation of the Rule

To understand why the divisibility rule for 12 works, let’s consider the factors of 12. The factors of 12 are 1, 2, 3, 4, 6, and 12. Since 12 is divisible by 2 and 3, any number that is divisible by 12 must also be divisible by 2 and 3.

Now, let’s look at the digits of a number. Suppose we have a number abcdef, where a, b, c, d, e, and f represent digits. By multiplying and adding these digits in a specific way, we can determine whether the number is divisible by 3. The rule is that if the sum of the digits is divisible by 3, then the entire number is also divisible by 3.

Similarly, to determine whether a number is divisible by 2, we only need to look at the last digit. If the last digit is even, then the number is divisible by 2.

Therefore, if the last two digits of a number are divisible by 4 (i.e., they form an even number that is divisible by 4), then the entire number is divisible by 4. Similarly, if the last three digits of a number are divisible by 8 (i.e., they form an even number that is divisible by 8), then the entire number is divisible by 8.

By applying this logic to the factors of 12, we can see that if the last two digits of a number are divisible by 4 and the sum of the digits is divisible by 3, then the entire number is divisible by 12.

## Examples of the Divisibility Rule for 12

Let’s consider some examples to see how the divisibility rule for 12 works.

- Is 3648 divisible by 12?
- The last two digits of 3648 are 48, which is divisible by 4.
- The sum of the digits is 3 + 6 + 4 + 8 = 21, which is divisible by 3.
- Therefore, 3648 is divisible by 12.

- Is 25128 divisible by 12?
- The last two digits of 25128 are 28, which is divisible by 4.
- The sum of the digits is 2 + 5 + 1 + 2 + 8 = 18, which is divisible by 3.
- Therefore, 25128 is divisible by 12.

- Is 7296 divisible by 12?
- The last two digits of 7296 are 96, which is divisible by 4.
- The sum of the digits is 7 + 2 + 9 + 6 = 24, which is divisible by 3.
- Therefore, 7296 is divisible by 12.

From the above examples, we can see that the divisibility rule for 12 is a useful tool for quickly determining whether a number is divisible by 12 or not.

## Why is the Divisibility Rule for 12 Important?

The divisibility rule for 12 is important because it can save time when performing mathematical calculations. For example, when working with fractions, we need to find a common denominator before we can add or subtract them. If we know that the denominators are divisible by 12, we can simplify the calculations by using the divisibility rule for 12 to find a common denominator quickly.

### What are the factors of 12?

The factors of 12 are 1, 2, 3, 4, 6, and 12.

### Can the divisibility rule for 12 be applied to all numbers?

No, the divisibility rule for 12 only applies to numbers that are multiples of 12.

### How can the divisibility rule for 12 be used in fractions?

If the denominators of two fractions are divisible by 12, we can simplify the fractions by using the divisibility rule for 12 to find a common denominator quickly.

### Can the divisibility rule for 12 be used to check if a number is prime?

No, the divisibility rule for 12 only checks whether a number is divisible by 12, not whether it is prime.

### Are there any other divisibility rules?

Yes, there are divisibility rules for other numbers, such as 2, 3, 4, 5, 6, 8, and 9.

## Conclusion

The divisibility rule for 12 is a simple and useful tool for quickly determining whether a number is divisible by 12 or not. By examining the last two digits of the number and checking if they are divisible by 4 and the sum of the digits is divisible by 3, we can determine whether the entire number is divisible by 12 or not. The rule is important because it can save time when performing mathematical calculations, especially when working with fractions.