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Chapter 3: Problem 18

Find the LCM of each set of numbers. $$ 18,20 $$

Short Answer

Expert verified
The LCM of 18 and 20 is 180.

Step by step solution

01

Prime Factorization

First, find the prime factorization for each number. For 18: - 18 = 2 脳 3^2 For 20: - 20 = 2^2 脳 5
02

Identify the Highest Powers

Next, identify the highest power of each prime number that appears in the factorizations. - The highest power of 2: 2^2 (from 20) - The highest power of 3: 3^2 (from 18) - The highest power of 5: 5 (from 20)
03

Multiply the Highest Powers

Finally, multiply these highest powers together to get the LCM. - LCM: 2^2 脳 3^2 脳 5 = 4 脳 9 脳 5 = 180

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Key Concepts

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

Prime Factorization
The first step in finding the Least Common Multiple (LCM) of numbers is breaking them down into their simplest prime factors. Prime factorization is expressing a number as the product of its prime numbers. Prime numbers are numbers that are only divisible by 1 and themselves, such as 2, 3, 5, 7, etc. For example, let's take 18 and 20:
  • 18 can be broken down into 2 and 3. Hence, 18 = 2 脳 3^2.
  • 20 can be broken down into 2 and 5. Thus, 20 = 2^2 脳 5.
Writing numbers as products of prime factors lays the groundwork for the next essential steps in finding the LCM.
Highest Powers
Once we have the prime factorization, the next step is to determine the highest powers of each prime that appear in any of the factorizations. This is crucial as the LCM must include every prime factor, raised to the greatest power that appears in any one of the factorizations. For instance:
  • The prime factor 2 appears as 2^1 in 18 and 2^2 in 20. So, the highest power of 2 to consider is 2^2.
  • The prime factor 3 appears as 3^2 in 18 but is absent in 20. So, we take 3^2 as the highest power of 3.
  • The prime factor 5 appears as 5^1 in 20 and does not appear in 18. Hence, the highest power is 5.
Comparing these factorizations ensures that our final LCM will be the smallest common multiple that includes all these prime numbers.
Multiplication of Primes
Now that we have selected the highest powers, the final step is to multiply these together to get the LCM. This step combines all the distinct prime factors at their highest powers. For our example, we'll multiply:
The highest power of 2: 2^2 = 4
The highest power of 3: 3^2 = 9
The highest power of 5: 5
Then, LCM = 4 脳 9 脳 5
First, 4 脳 9 = 36, and then 36 脳 5 = 180. Therefore, the LCM of 18 and 20 is 180. The process of multiplying these highest powers ensures that all factors are included, making the resultant multiple the least common one for all the numbers provided.

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