Problem 55 - Algebra A Combined Function

Question

Objective C Find the LCM of each list of numbers. See Examples 7 through 9 . 8,24,48

Step-by-Step Solution

Answer

The LCM of 8, 24, and 48 is 48.

Step 1: Prime Factorization

First, find the prime factorization of each number. For 8, the prime factorization is \( 2^3 \). For 24, the prime factorization is \( 2^3 \times 3^1 \). For 48, the prime factorization is \( 2^4 \times 3^1 \).

Step 2: Identify Maximum Powers of Each Prime

In the prime factorizations, identify the maximum power for each prime number across all numbers. For 2, the maximum power is 4 (from 48), and for 3, the maximum power is 1 (from both 24 and 48).

Step 3: Calculate the LCM

Multiply the highest powers of all primes identified in the previous step. The LCM is calculated as \( 2^4 \times 3^1 = 16 \times 3 = 48 \).

Key Concepts

Prime Factorization

Understanding prime factorization is essential for finding the Least Common Multiple (LCM) of a set of numbers. Prime factorization involves breaking down a number into its prime number components—those that can’t be divided further except by 1 and themselves. For instance:

• For the number 8, the prime factors are 2, multiplied together three times: \( 2^3 \).
• The number 24 breaks down into \( 2^3 \times 3^1 \). This means three 2s and one 3.
• The number 48 breaks into \( 2^4 \times 3^1 \), involving four 2s and one 3.

Prime factorization reveals the foundation of each number, allowing us to compare and compute efficiently. Recognizing the highest powers of each prime across the numbers in our list will help us determine the least common multiple.

Mathematical Problem Solving

When solving for the LCM using prime factorization, the process is straightforward but requires attention to detail. The problem-solving steps ensure you understand and can apply the method efficiently:

• First, decompose all numbers into their prime factors. Each number should be expressed solely in terms of primes.
• Next, note the highest power of each prime factor present in the breakdowns. These are crucial for our calculations.
• Combine these maximum powers. This involves multiplication of the primes raised to their identified highest powers.

For our example, multiplying the maximum powers: \( 2^4 \) from 48 and \( 3^1 \) from both 24 and 48, results in the LCM, which calculates to 48. Good problem-solving is about precision and following a step-by-step approach.

Educational Step-by-Step Solutions

Educational step-by-step solutions play a vital role in understanding complex mathematical concepts such as finding the LCM. These solutions break down the exercise into manageable steps, ensuring clarity and comprehension.

• Step 1: Start by finding the prime factorization of each number, simplifying the task of comparing them.
• Step 2: Identify and compare the powers of each prime number used, focusing on the highest power of each prime.
• Step 3: Calculate the LCM by multiplying these highest powers, ensuring you only use the most pivotal values.

The structured approach ensures no component is overlooked and reinforces the intuitive understanding of mathematical relationships. By following each step, students can solve even the most challenging problems with confidence and accuracy.