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Chapter 3: Problem 8

What is the lowest common multiple of 6 and 8?

Short Answer

Expert verified
24

Step by step solution

01

- List the Prime Factors

Find the prime factors of each number. For 6: the prime factors are 2 and 3.For 8: the prime factors are 2, 2, and 2.
02

- Find the Highest Power of Each Prime Factor

Identify the highest powers of all prime numbers involved.For the number 6: 2¹ and 3¹.For the number 8: 2³.
03

- Multiply the Highest Powers of Prime Factors

Multiply the highest powers of each prime to get the lowest common multiple.This would be:\( 2^3 \times 3^1 \).Thus, the calculation is: 8 × 3 = 24.

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Key Concepts

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

Prime Factors
Prime factors are the building blocks of numbers. A prime factor is a prime number that can be multiplied to get the original number. For example, the prime factors of 6 are 2 and 3 because 2 x 3 = 6.

Similarly, the prime factors of 8 are three 2s (i.e., 2 x 2 x 2 = 8). Finding prime factors is crucial because it helps us break down numbers into their most basic components.

To find the prime factors, follow these steps:
  • Divide the number by the smallest prime number (usually 2) that goes into the number evenly.
  • Continue dividing the result by the smallest prime number until you can no longer divide evenly.
  • The prime numbers you used for division are the prime factors of your original number.
Multiplication
Multiplication is a fundamental math operation. When we multiply numbers, we are adding equal groups of a number together. For example, 4 × 3 means adding four groups of 3, which equals 12.

In the context of finding the Lowest Common Multiple (LCM), multiplication is used to combine the highest powers of all prime factors involved. This step ensures that the resulting number is a multiple of both original numbers. For example, if you have the highest powers of the prime numbers 2 and 3, you multiply them as follows:
The highest power of 2 is 2³ (which is 8) and the highest power of 3 is 3¹ (which is 3).

To find the LCM, you would multiply 8 by 3, giving you 24.
LCM Calculation
LCM, or Lowest Common Multiple, is the smallest number that is a multiple of two or more numbers. To calculate the LCM, follow these steps:

  • Find the prime factors of each number. For example, 6 has prime factors 2 and 3, while 8 has prime factors 2, 2, and 2.
  • Identify the highest power of each prime factor. For the numbers 6 and 8, the highest power of 2 is 2³, and for 3 it’s 3¹.
  • Multiply these highest powers together. In this example, it’s 2³ (which is 8) multiplied by 3¹ (which is 3).

By following these steps: The calculation for LCM of 6 and 8 is 2³ x 3¹ = 8 x 3 = 24. Thus, the LCM of 6 and 8 is 24.

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