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Chapter 3: Problem 42

Find the LCM of each set of numbers. $$ 56,72 $$

Short Answer

Expert verified
504

Step by step solution

01

- Prime Factorization

Find the prime factorization of each number. The prime factorization of 56 is: \[ 56 = 2^3 \times 7 \] The prime factorization of 72 is: \[ 72 = 2^3 \times 3^2 \]
02

- Identify the Highest Powers

Identify the highest powers of each prime number common to the factorizations. - For 2, the highest power is \(2^3\). - For 3, the highest power is \(3^2\). - For 7, the highest power is \(7\).
03

- Calculate the LCM

Multiply these highest powers together to find the Least Common Multiple (LCM). \[ LCM = 2^3 \times 3^2 \times 7 \] Calculate the result: \[ 2^3 = 8 \] \[ 3^2 = 9 \] \[ LCM = 8 \times 9 \times 7 = 504 \]

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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 the prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, 2, 3, 5, and 7 are prime numbers.
To find the prime factorization of a number, you divide the number by the smallest prime number until you cannot divide any further. For instance, to find the prime factorization of 56, you divide:
  • 56 梅 2 = 28
  • 28 梅 2 = 14
  • 14 梅 2 = 7
  • 7 梅 7 = 1
This gives us \( 56 = 2^3 \times 7 \). Similarly, prime factorizing 72 gives us \( 72 = 2^3 \times 3^2\). This essential step is crucial for calculating the Least Common Multiple (LCM).
least common multiple
The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. To find the LCM using prime factorization, you need to:
  • Find the prime factorization of each number.
  • Identify the highest powers of each prime number.
  • Multiply these highest powers together.
For 56 and 72, the highest powers of each prime factor found in their prime factorizations are:
For 2: \(2^3\)
For 3: \(3^2\)
For 7: \(7\)
Multiplying these together gives: \( LCM = 2^3 \times 3^2 \times 7 = 504\).
This process ensures we find the smallest number that both original numbers can divide into without leaving a remainder.
mathematical problem solving
Mathematical problem solving involves a series of steps to understand and solve a problem effectively. The steps often include:
  • Understanding the problem: Clearly define what is being asked.
  • Devising a plan: Determine the method needed to solve the problem (like using prime factorization for LCM).
  • Carrying out the plan: Execute the mathematical operations as devised.
  • Reviewing the solution: Check if the solution is correct and makes sense.
Following these steps helps to systematically break down more complex problems into manageable parts, ensuring accuracy and understanding in finding solutions.
number theory
Number theory is a branch of mathematics focused on the properties and relationships of numbers, particularly integers. Key concepts in number theory include:
  • Prime numbers: Numbers greater than 1 that are divisible only by 1 and themselves.
  • Factorization: Expressing numbers as a product of primes and other integers.
  • LCM and GCD (Greatest Common Divisor): Finding common multiples and divisors of numbers.
In this problem, number theory principles are applied to find the LCM of 56 and 72 using their prime factorizations. This knowledge illuminates how numbers interact and helps solve practical problems involving divisibility, common multiples, and more.

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Most popular questions from this chapter

Simplify. $$ \frac{3}{4} \div \frac{1}{2} \cdot\left(\frac{8}{9}-\frac{2}{3}\right) $$

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