Common Multiples Of 48 And 60

Finding the Common Multiples of 48 and 60: A Comprehensive Guide

Finding the common multiples of two numbers might seem daunting, but with the right approach, it's a straightforward process. This article will guide you through the steps of finding the common multiples of 48 and 60, explaining the concepts and providing practical methods for solving similar problems. Understanding common multiples is fundamental in various mathematical applications, including fractions, ratios, and even more advanced concepts.

What are Multiples and Common Multiples?

A multiple of a number is the result of multiplying that number by any integer (whole number). For example, multiples of 48 include 48 (48 x 1), 96 (48 x 2), 144 (48 x 3), and so on. Similarly, multiples of 60 include 60 (60 x 1), 120 (60 x 2), 180 (60 x 3), and so forth.

Common multiples, as the name suggests, are multiples that are shared by two or more numbers. In our case, we're looking for numbers that appear in both the list of multiples of 48 and the list of multiples of 60.

Method 1: Listing Multiples

This method is straightforward but can become tedious for larger numbers. Let's list the first few multiples of 48 and 60:

Multiples of 48: 48, 96, 144, 192, 240, 288, 336, 384, 432, 480, 528...

Multiples of 60: 60, 120, 180, 240, 300, 360, 420, 480, 540...

By comparing the two lists, we can immediately identify some common multiples: 240 and 480. You'll notice that this method becomes less efficient as you need to list more multiples to find common ones.

Method 2: Using Prime Factorization (More Efficient)

This method is far more efficient, especially for larger numbers. It involves finding the prime factorization of each number.

• Prime Factorization of 48: 2 x 2 x 2 x 2 x 3 = 2⁴ x 3
• Prime Factorization of 60: 2 x 2 x 3 x 5 = 2² x 3 x 5

To find the least common multiple (LCM), we take the highest power of each prime factor present in either factorization:

LCM(48, 60) = 2⁴ x 3 x 5 = 16 x 3 x 5 = 240

The LCM is the smallest common multiple. All other common multiples are multiples of the LCM. Therefore, the common multiples of 48 and 60 are: 240, 480, 720, 960, and so on. These are all multiples of 240.

Method 3: Using the Formula (LCM and GCD)

You can also use the formula that relates the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) of two numbers:

LCM(a, b) = (a x b) / GCD(a, b)

First, find the GCD of 48 and 60 using the Euclidean algorithm or prime factorization. The GCD(48, 60) = 12.

Then, apply the formula:

LCM(48, 60) = (48 x 60) / 12 = 2880 / 12 = 240

Again, the LCM is 240, and all common multiples are multiples of 240.

Conclusion:

Finding common multiples is crucial for solving various mathematical problems. While listing multiples is a simple approach, prime factorization provides a more efficient and systematic method, particularly when dealing with larger numbers. Understanding these methods empowers you to confidently tackle problems involving common multiples and related concepts. Remember that all common multiples are multiples of the least common multiple (LCM).