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Chapter 3: Problem 1

Find the LCM of each set of numbers. $$ 2,4 $$

Short Answer

Expert verified
The LCM is 4.

Step by step solution

01

List the prime factors of each number

Prime factorize each number in the set. - The prime factors of 2 are: \[2\] - The prime factors of 4 are: \[2^2\]
02

Identify the highest powers of each prime factor

For each prime factor found in the step above, note the highest power. - The highest power of 2 is \[2^2\]
03

Multiply the highest powers of each prime factor

Multiply the highest powers of all the prime factors to find the LCM. - Therefore, \[2^2 = 4\]

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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 its basic building blocks, which are prime numbers. Every number can be expressed as a product of prime numbers. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

Let's illustrate this with the numbers 2 and 4.
  • For the number 2, since it is already a prime number, its prime factorization is simply \[2\].
  • For the number 4, we break it down into prime factors by dividing it by the smallest prime number (2). So, \[4 = 2 \times 2 = 2^2\].
Understanding prime factorization is crucial because it helps in solving problems related to the least common multiple (LCM) and greatest common divisor (GCD). It is a fundamental concept in various areas of mathematics, including algebra and number theory.
LCM calculation
The LCM, or Least Common Multiple, of two or more numbers is the smallest number that is a multiple of each of the numbers. To calculate the LCM using prime factorization, follow these steps:

First, list the prime factors of each number. As previously mentioned:
  • The prime factors of 2 are \[2\]
  • The prime factors of 4 are \[2^2\]
Next, identify the highest power of each prime factor found in the step above. In our example:
  • The highest power of 2 is \[2^2\]
Finally, multiply the highest powers of each prime factor:
  • Here, the LCM is \[2^2 = 4\]
Therefore, the LCM of 2 and 4 is 4. This method ensures that you include all necessary factors at their maximum powers, making it a reliable way to find the LCM.
mathematics education
Understanding concepts like prime factorization and LCM is fundamental in mathematics education. These topics form the building blocks for more advanced mathematical concepts. Familiarity with these concepts enhances problem-solving skills and mathematical reasoning.

When teaching or learning these concepts, consider the following tips:
  • Use visual aids, like factor trees, to break down numbers into prime factors.
  • Practice with a variety of numbers to build confidence and proficiency.
  • Relate the concepts to real-life scenarios, such as finding the LCM of time intervals.
Interactive exercises and step-by-step problem solving, as shown in our example, help reinforce understanding. Encouraging questions and offering clear, concise explanations can make learning math an enjoyable and rewarding experience.

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Most popular questions from this chapter

Estimate each of the following by estimating each mixed numeral as a whole number or as a mixed numeral where the fraction part is \(\frac{1}{2}\) and by estimating each fraction as \(0, \frac{1}{2},\) or 1 . $$ 12 \frac{5}{6} $$

Simplify. $$ \frac{\frac{3}{5}-\frac{1}{2}}{\frac{1}{2}-\frac{2}{5}} $$

Simplify. Find the prime factorization of \(150 .[2.1 \mathrm{~d}]\)

A tile \(\frac{5}{8}\) in. thick is glued to a board \(\frac{7}{8}\) in. thick. The glue is \(\frac{3}{32}\) in. thick. How thick is the result?

Simplify. Classify the given numbers as prime, composite, or neither. \([2.1 \mathrm{c}]\) $$ 1,5,7,9,14,23,43 $$
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