Divisibility rules are an essential concept in mathematics that helps us determine whether a number is divisible by another number without performing long division or factorization. In this article, we will discuss the divisibility rule for 15 and its application in solving mathematical problems.

## What is the Divisibility Rule for 15?

The divisibility rule for 15 states that a number is divisible by 15 if it is divisible by both 3 and 5. In other words, a number is divisible by 15 if the sum of its digits is divisible by 3, and the number ends with a 5 or a 0.

**For example**, let’s consider the number 495. The sum of its digits is 4 + 9 + 5 = 18, which is divisible by 3. Also, the number ends with a 5, which means 495 is divisible by 15.

## Understanding the Mathematical Concept of Divisibility Rule for 15

To understand the mathematical concept of the divisibility rule for 15, we need to understand the properties of numbers that make them divisible by 3 and 5.

### Divisibility Rule for 3

A number is divisible by 3 if the sum of its digits is divisible by 3. For example, the number 123 is divisible by 3 because 1 + 2 + 3 = 6, which is divisible by 3.

### Divisibility Rule for 5

A number is divisible by 5 if it ends with a 5 or a 0. For example, the number 150 is divisible by 5 because it ends with a 0.

### Divisibility Rule for 15

A number is divisible by 15 if it is divisible by both 3 and 5. This means that the number must meet the conditions of both the divisibility rules for 3 and 5.

## Application of Divisibility Rule for 15

The divisibility rule for 15 is useful in many mathematical problems, especially in simplifying complex calculations. Here are some examples of how to apply the rule in solving mathematical problems.

### Example 1: Is 105 divisible by 15?

To determine whether 105 is divisible by 15, we need to check if it is divisible by both 3 and 5.

The sum of the digits of 105 is 1 + 0 + 5 = 6, which is divisible by 3. Also, the number ends with a 5, which means 105 is divisible by 5.

Since 105 is divisible by both 3 and 5, it is divisible by 15.

### Example 2: Find the smallest positive integer that is divisible by both 3 and 5.

To find the smallest positive integer that is divisible by both 3 and 5, we need to find the least common multiple (LCM) of 3 and 5.

The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, …

The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, …

The least common multiple of 3 and 5 is 15. Therefore, the smallest positive integer that is divisible by both 3 and 5 is 15.

### Example 3: Simplify the fraction 45/15.

To simplify the fraction 45/15, we can use the divisibility rule for 15.

Since both 45 and 15 are divisible by 15, we can simplify the fraction by dividing both the numerator and the denominator by 15.

45/15 = (15 x 3)/(15 x 1) = 3/1 = 3

**Therefore, 45/15 simplifies to 3.**

## FAQs

### What is the divisibility rule for 3?

The divisibility rule for 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3.

### What is the divisibility rule for 5?

The divisibility rule for 5 states that a number is divisible by 5 if it ends with a 5 or a 0.

### How do you apply the divisibility rule for 15?

To apply the divisibility rule for 15, you need to check if the number is divisible by both 3 and 5. This means that the sum of the digits of the number must be divisible by 3, and the number must end with a 5 or a 0.

### What is the least common multiple of 3 and 5?

The least common multiple of 3 and 5 is 15.

### How do you simplify a fraction using the divisibility rule for 15?

To simplify a fraction using the divisibility rule for 15, you need to check if both the numerator and denominator are divisible by 15. If they are, divide both by 15 to simplify the fraction.

## Conclusion

The divisibility rule for 15 is a mathematical concept that helps us determine whether a number is divisible by 15 without performing long division or factorization. The rule states that a number is divisible by 15 if it is divisible by both 3 and 5. This means that the sum of the digits of the number must be divisible by 3, and the number must end with a 5 or a 0. The divisibility rule for 15 is useful in simplifying complex calculations and solving mathematical problems.