Multiples and Lowest Common Multiple

Introduction

Multiples and their smallest common equivalent, known as the Lowest Common Multiple (LCM), play crucial roles in a range of applications in the field of mathematics. In addition to helping you solve math problems, understanding these ideas can help you understand the underlying ideas that underpin many facets of daily life.

Multiple

A multiple of a number is a multiplication of its factor pairs. A factor pair is a set of two numbers that, when multiplied together, equal the original number. For example, the factor pairs of 12 are (1, 12), (2, 6), and (3, 4).

Therefore, any multiple of 12 can be written as the product of two of its factor pairs. For example, 24 is a multiple of 12 because it can be written as (2, 12).

The converse is also true. Any product of two factor pairs of a number is a multiple of that number. For example, (3, 4) is a factor pair of 12, so 12 is a multiple of (3, 4).

Now, let’s explore the concept of multiples through a simple example. Suppose we want to find the first five multiples of 5. Multiples are numbers that result from multiplying a given number by integers in sequence.

1. 5 x 1 = 5 (Here, 5 is the first multiple of 5)
2. 5 x 2 = 10 (10 is the second multiple of 5)
3. 5 x 3 = 15 (15 is the third multiple of 5)
4. 5 x 4 = 20 (20 is the fourth multiple of 5)
5. 5 x 5 = 25 (25 is the fifth multiple of 5)

In this sequence, we start with 5, and then by multiplying it by consecutive integers (1, 2, 3, 4, and 5), we obtain the first five multiples of 5.

The multiples of any nonzero integer are indeed infinite

In the previous example, I provided the first five multiples of 5, but in reality, there is no end to the list of multiples of 5. You can keep multiplying 5 by consecutive positive integers, and you will continue to get new multiples of 5 indefinitely. This concept applies not only to 5 but to any nonzero integer.

So, to clarify, the multiples of any number extend infinitely in both positive and negative directions, forming an infinite sequence. For example, the multiples of 5 include 5, 10, 15, 20, 25, 30, and so on, as well as their negative counterparts like -5, -10, -15, -20, -25, -30, and so forth.

Lowest / Least Common Multiple ( LCM )

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of the numbers

For example, the LCM of 2 and 3

Multiple of 2 = 2, 4, 6, 8, 10, 12, 14

Multiple of 3 = 3, 6, 9, 12, 15, 18, 21

Here, 6 is lowest number that is common on both the multiples of 2 & 3

There are two main methods for finding the LCM of two or more numbers:

1. Prime Factorization Method
2. Greatest Common Divisor (GCD) Method
3. List Method

Method 1 : LCM by Prime Factorization Method

The prime factorization method is a systematic way to find the LCM. It involves breaking down each number into its prime factors and then constructing the LCM using the highest powers of these prime factors.

Example 1: Find the LCM of 12 and 18

Step 1: Prime factorization of 12 and 18

Prime Factorization of 12 = 2 x 6 = 2 X 2 X 3 = 2² X 3¹

Prime Factorization of 18 = 2 X 9 = 2 X 3 X 3 = 2¹ X 3²

Step 2: Take the highest powers of prime factors

• LCM = 2² X 3² = 4 X 9 = 36

So, the LCM of 12 and 18 is 36.

Method 2: Using the List Method

For smaller numbers, you can use the list method, where you list the multiples of each number until you find a common multiple.

Example 3: Find the LCM of 3 and 5

Multiples of 3: 3, 6, 9, 12, 15, … Multiples of 5: 5, 10, 15, 20, 25, …

The common multiple is 15, so LCM = 15.

Method 3: LCM Using the GCD / HCF

Another method to find the LCM is by first determining the Highest Common Factor / Greatest Common Divisor (GCD) of the given numbers. Then, you can use a simple formula to find the LCM.

Example 2: Find the LCM of 16 and 24

Step 1: Find the GCD/ HCF of 16 and 24

Factor of 16 = 1, 2, 4, 8, 16

Factor of 24 = 1, 2, 3, 4, 6, 8, 12

Therefore, HCF (16, 24) = 8

Step 2: Use the formula LCM X HCF = Number1 X Number2

LCM X 8 = 16 X 24

LCM = 16 x 24/8

= 48

Therefore, The LCM of 16 and 24 is 48.

Applications of LCM in Real Life

Understanding LCM involves more than just figuring out mathematical puzzles; it also has real-world uses. LCM supports effective work scheduling, event coordination, and process optimization in a variety of industries, including engineering, computer science, and even daily living.

Conclusion

You can improve your mathematical aptitude and problem-solving skills by becoming an expert at determining the LCM. LCM is a fundamental idea that is essential to both mathematics and everyday life, regardless of the approach you use (prime factorization, GCD, or list). The next time you come across multiple numbers, remember to calculate their LCM because it might just hold the key to finding the answer.