Understanding the Least Common Multiple (LCM) of 5, 10, and 10: A Step-by-Step Guide

Mathematics, specifically number theory, often involves finding the least common multiple (LCM) of a set of numbers. In this article, we will explore how to find the LCM of 5, 10, and 10 using different methods. Understanding the LCM is crucial in various mathematical operations, such as adding or subtracting fractions with different denominators.

What is the LCM of 5, 10, and 10?

The LCM of a set of numbers is the smallest positive integer that is divisible by each of the numbers in the set. For the numbers 5, 10, and 10, the LCM is 10. Here’s a step-by-step explanation of how to find this:

Step 1: List the Multiples

To find the LCM of 5, 10, and 10, we start by listing the multiples of each number:

Multiples of 5: 5, 10, 15, 20, 25, 30, ... Multiples of 10: 10, 20, 30, 40, ... Multiples of 10 (repeated): 10, 20, 30, 40, ...

As you can see, 10 is the smallest number that appears in all three lists of multiples. Thus, the LCM of 5, 10, and 10 is 10.

Step 2: Prime Factorization Method

A more systematic way to find the LCM is through prime factorization:

Prime factorization of 5: 5 5^1 Prime factorization of 10: 10 2^1 × 5^1 The highest power of each prime factor: 2^1 and 5^1

Multiply the highest powers of each prime factor together:

2^1 × 5^1 2 × 5 10

Therefore, the LCM of 5, 10, and 10 is 10.

Explanation of the LCM

The LCM of two or more numbers is the smallest number that is a multiple of each of the numbers. It represents the smallest amount that can be evenly divided by all the numbers in the set without leaving a remainder. For example, the LCM of 5, 10, and 10 is 10 because 10 is the smallest number that can be divided by 5, 10, and 10 without any remainder.

Visual Representation

Let's break down the numbers into their prime factors:

Px 5

Qx 10 2 x 5

Rx 10 2 x 5

The common factor is 5, which is included once, and the unique factors are 2, which are also included once.

Therefore, the LCM 2 x 5 10

Conclusion

The least common multiple (LCM) of 5, 10, and 10 is 10. This holds true regardless of whether you list the multiples or use the prime factorization method.

Understanding the LCM is essential for advanced mathematical operations and problem-solving. If you have any further questions or need additional clarification, feel free to ask. Click the link below to learn more about LCM and other mathematical concepts.