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Chapter 4: Problem 24

Find the LCM. \(4,6,\) and 14

Short Answer

Expert verified
The LCM is 84.

Step by step solution

01

- List the Prime Factors

Determine and list the prime factors of each number:- 4: 4 = 2 脳 2 - 6: 6 = 2 脳 3 - 14: 14 = 2 脳 7
02

- Identify the Highest Power of Prime Factors

Find the highest power of each prime number that appears in the factorizations:- For 2: The highest power is 2 (from 4, which is 2^2)- For 3: The highest power is 3 (from 6)- For 7: The highest power is 7 (from 14)
03

- Calculate the LCM

Multiply the highest powers of all the prime factors found in step 2:LCM = 2^2 脳 3 脳 7 = 4 脳 3 脳 7First calculate: 4 脳 3 = 12Then, 12 脳 7 = 84

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Key Concepts

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

Prime Factorization
Prime factorization is the process of breaking down a number into prime numbers that, when multiplied together, give the original number. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves (e.g., 2, 3, 5, 7).

To prime factorize a number:
  • Start with the smallest prime number (2) and divide the number by it.
  • If the number is divisible by 2, divide and write down 2 as a factor. Continue this process until the result is no longer divisible by 2.
  • Move on to the next smallest prime number (3) and repeat the process.
  • Continue until the number is completely factorized into prime numbers.
For example, the prime factorization of 4 is 2 脳 2, the prime factorization of 6 is 2 脳 3, and the prime factorization of 14 is 2 脳 7.
Highest Power of Prime Factors
When finding the Least Common Multiple (LCM) of a set of numbers, it's important to consider the highest power of all prime factors that appear in the prime factorizations.

Here's how to identify the highest power of prime factors:
  • List all the prime factors for each number.
  • Compare the exponents of each prime factor across all numbers.
  • Select the highest exponent for each prime factor.
For example, looking at the numbers 4, 6, and 14:
  • Prime factors of 4 (2 脳 2): The highest power of 2 is 2^2.
  • Prime factors of 6 (2 脳 3): The highest power of 3 is 3^1.
  • Prime factors of 14 (2 脳 7): The highest power of 7 is 7^1.
By choosing the highest power of each prime factor, it ensures the multiples of each number are covered in the LCM.
Multiplying Factors
After identifying the highest power of each prime factor, the next step is to multiply these factors together to find the LCM.

To multiply factors:
  • Write down the highest power of each prime factor found in the previous step.
  • Multiply these prime factors step-by-step, simplifying as needed.
For instance, to find the LCM of 4, 6, and 14:
  • The prime factors are 2^2, 3, and 7.
  • First, multiply 2^2 and 3: 2^2 脳 3 = 4 脳 3 = 12.
  • Then, multiply the result by 7: 12 脳 7 = 84.
Thus, the LCM of 4, 6, and 14 is 84. This means 84 is the smallest number that is a multiple of all three numbers.

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