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Chapter 10: Problem 69

Find the LCM. $$ 18,24 $$

Short Answer

Expert verified
The LCM of 18 and 24 is 72.

Step by step solution

01

- Find the Prime Factors

First, find the prime factors of each number. The prime factors of 18 are 2 and 3 because 18 = 2 脳 3虏. The prime factors of 24 are 2 and 3 because 24 = 2鲁 脳 3.
02

- Identify the Highest Powers

Next, identify the highest powers of the prime factors found in the previous step. For 2, the highest power found is 2鲁 (from 24). For 3, the highest power found is 3虏 (from 18).
03

- Multiply the Highest Powers

Finally, multiply these highest powers together to find the LCM. The LCM is 2鲁 脳 3虏.
04

- Calculate the LCM

Calculate the result of 2鲁 脳 3虏. This equals 8 脳 9, which equals 72.

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Key Concepts

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

Prime Factors
Understanding prime factors is crucial when finding the Least Common Multiple (LCM). Prime factors are the building blocks of all numbers, as they are the prime numbers that multiply together to create the original number. To break down a number into its prime factors, you start by dividing it by the smallest prime number, which is 2, and continue dividing by prime numbers (3, 5, 7, etc.) until you can no longer divide evenly.

For example:
  • The prime factors of 18 are 2 and 3, because 18 = 2 脳 3虏.
  • The prime factors of 24 are 2 and 3, because 24 = 2鲁 脳 3.


Getting comfortable with prime factorization helps to streamline the process of finding the LCM and understanding how numbers relate to each other.
Highest Powers
Once you have identified the prime factors, the next step is to determine the highest powers of these factors. This is important because the LCM must include the highest powers of all prime factors present in any of the numbers you're comparing.

For instance:
  • The highest power of 2 between 18 and 24 is 2鲁, because 24 = 2鲁 脳 3.
  • The highest power of 3 between 18 and 24 is 3虏, because 18 = 2 脳 3虏.


By collecting the highest powers, we ensure that the LCM is a number that can be evenly divided by all original numbers without leaving a remainder.
Multiplication of Factors
The final step in finding the Least Common Multiple is multiplying the highest powers of the prime factors together. This step gives us the smallest number that is a multiple of each original number.

Following the example from above:
  • The highest power of 2 is 2鲁 (which is 8), and the highest power of 3 is 3虏 (which is 9).
  • To find the LCM, multiply these together: 8 脳 9.


    • This results in 72, so the LCM of 18 and 24 is 72.

    By understanding and applying the concepts of prime factorization and highest powers, finding the LCM becomes a systematic process that can be applied to any set of integers.

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