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Chapter 3: Problem 30

Find the LCM of each set of numbers. $$ 8,16,12 $$

Short Answer

Expert verified
The LCM of 8, 16, and 12 is 48.

Step by step solution

01

Prime Factorization

Find the prime factorization of each number: 8: 8 = 2 脳 2 脳 2 = 2^3 16: 16 = 2 脳 2 脳 2 脳 2 = 2^4 12: 12 = 2 脳 2 脳 3 = 2^2 脳 3^1
02

Identify the Highest Powers

Identify the highest power of each prime factor: The highest power of 2 is 2^4 (from 16). The highest power of 3 is 3^1 (from 12).
03

Multiply the Highest Powers Together

Multiply the highest powers of each prime factor together: LCM = 2^4 脳 3^1 = 16 脳 3 = 48

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization
Understanding prime factorization is key to solving LCM problems. Prime factorization is the process of breaking down a number into prime numbers that multiply together to give the original number. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves, like 2, 3, 5, 7, etc.

For example:
To find the prime factors of 8, we divide 8 by the smallest prime number, 2, resulting in 4. We continue dividing by 2 until we can't anymore. This gives us 8 = 2 脳 2 脳 2 or 2^3.

Similarly, we find that 16 becomes 2^4, and 12 becomes 2^2 脳 3^1. Breaking down numbers into their prime factors makes it easier to work with them in calculations like finding the LCM.
Lowest Common Multiple
The Lowest Common Multiple, or LCM, of a set of numbers is the smallest number that is a multiple of all numbers in the set. Knowing how to find the LCM is extremely useful in various math problems, including those involving fractions and ratios.

The LCM is found by taking the highest power of each prime factor from the numbers you're comparing. Here鈥檚 how:
  • First, perform prime factorization for each number.
  • Identify the highest powers of these prime factors.
  • Multiply these highest powers together.
For example, with 8 ( = 2^3), 16 ( = 2^4), and 12 ( = 2^2 脳 3^1), the highest power of 2 is 2^4 and the highest power of 3 is 3^1. Multiplying these together, we get the LCM as 2^4 脳 3^1 = 48.
Mathematics Education
In mathematics education, understanding concepts like prime factorization and LCM is essential. These topics frequently appear in algebra, number theory, and problem-solving.

Learning prime factorization helps in simplifying complex equations, while understanding LCM aids in solving problems involving comparisons and units.

By mastering these basics, students build a strong foundation for advanced math topics. Interactive exercises, practical applications, and step-by-step guides are effective ways to teach these concepts, ensuring students not only memorize them but also understand their real-world significance.

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Most popular questions from this chapter

Find the LCM of each set of numbers. $$ 18,27 $$

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