Learn how to find the LCM of 3 and 10, the fundamental concept in mathematics. Discover what the least common multiple is and how to calculate it.

## Introduction

In mathematics, the concept of Least Common Multiple (LCM) plays an essential role in solving various problems related to fractions, algebraic equations, and arithmetic operations. It is one of the fundamental topics taught in elementary mathematics. The LCM of two or more numbers is the smallest number that is a multiple of all the given numbers. In this article, we will discuss how to find the least common multiple of 3 and 10.

## Finding the Multiples of 3 and 10

To find the LCM of 3 and 10, we first need to understand what multiples are. Multiples are the products obtained by multiplying any number by another whole number. For example, the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, and so on. Similarly, the multiples of 10 are 10, 20, 30, 40, 50, 60, 70, and so on.

To find the first few multiples of a number, you can simply multiply the given number by the first few natural numbers. For example, to find the first four multiples of 3, we can multiply 3 by 1, 2, 3, and 4, which gives us 3, 6, 9, and 12. Similarly, the first four multiples of 10 can be found by multiplying 10 by 1, 2, 3, and 4, which gives us 10, 20, 30, and 40.

By listing the multiples of 3 and 10, we can observe that they have some common multiples. These common multiples are the numbers that are multiples of both 3 and 10. The common multiples of 3 and 10 are 30, 60, 90, 120, and so on. However, not all multiples of 3 and 10 are common multiples. For example, 6 is a multiple of 3 but not a multiple of 10, so it is not a common multiple. Similarly, 20 is a multiple of 10 but not a multiple of 3, so it is not a common multiple.

## Identifying the Common Multiples

Once we have identified the multiples of 3 and 10, the next step is to find the common multiples. Common multiples are those multiples that are divisible by both 3 and 10. In other words, they are the common elements in the list of multiples of 3 and 10.

The common multiples of 3 and 10 are 30, 60, 90, 120, and so on. We can observe that these common multiples are obtained by multiplying 3 and 10 by the same integer. For example, 30 is the product of 3 and 10, 60 is the product of 3 and 20, and so on.

## Identifying the Least Common Multiple

Now that we have identified the common multiples of 3 and 10, the next step is to find the least common multiple. The least common multiple is the smallest common multiple of the given numbers. In other words, it is the smallest number that is divisible by both 3 and 10.

To find the LCM of 3 and 10, we can use the common multiples we have identified. We can observe that the smallest common multiple of 3 and 10 is 30. This is because 30 is the first common multiple of 3 and 10, and it is the smallest number that is divisible by both 3 and 10.

We can also find the LCM of 3 and 10 by using the prime factorization method. To do this, we first need to factorize each number into its prime factors. The prime factors of 3 are 3, and the prime factors of 10 are 2 and 5.

Next, we can write the prime factors of each number in the form of a product of primes. For example, 3 can be written as 3^1, and 10 can be written as 2^1 x 5^1.

To find the LCM, we take the highest power of each prime factor that appears in either factorization. In this case, the highest power of 2 is 2^1, the highest power of 3 is 3^1, and the highest power of 5 is 5^1.

Multiplying these three factors gives us the LCM of 3 and 10, which is 2^1 x 3^1 x 5^1 = 30. Therefore, we can conclude that the LCM of 3 and 10 is 30.

## Applying the Concept of Least Common Multiple

### Example Problem on Finding the LCM of 3 and 10

Let’s consider the problem of finding the LCM of 3 and 10. To find the LCM of two numbers, we can use the following method:

- List the multiples of each number until you find a common multiple.
- The smallest common multiple is the LCM.

Using this method, we can list the multiples of 3 and 10 as follows:

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, …

Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, …

From the above lists, we can see that the smallest common multiple of 3 and 10 is 30. Therefore, the LCM of 3 and 10 is 30.

### Step-by-Step Solution to the Example Problem

- List the multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, …
- List the multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, …
- Identify the smallest number that appears in both lists: 30.
- Therefore, the LCM of 3 and 10 is 30.

## Conclusion

In conclusion, the least common multiple of two or more numbers is the smallest number that is a multiple of all the given numbers. It is an essential concept in mathematics that is used in solving various problems. By finding the LCM of two or more numbers, we can simplify fractions, solve algebraic equations, and perform arithmetic operations. It is important to understand the concept of LCM and know how to find it to excel in mathematics.