Problem 52 - Algebra A Combined Function

Question

Objective C Find the LCM of each list of numbers. See Examples 7 through 9 . 3,9,20

Step-by-Step Solution

Answer

The LCM of 3, 9, and 20 is 180.

Step 1: Prime Factorization

First, find the prime factorization of each number.- The prime factorization of 3 is \(3 = 3^1\).- The prime factorization of 9 is \(9 = 3^2\).- The prime factorization of 20 is \(20 = 2^2 \times 5^1\).

Step 2: Identify the Highest Powers

Identify the highest power of each prime number from the factorizations:- The highest power of 2 is \(2^2\) from 20.- The highest power of 3 is \(3^2\) from 9.- The highest power of 5 is \(5^1\) from 20.

Step 3: Multiply the Highest Powers

The LCM is found by multiplying together the highest powers of all primes involved.Calculate the LCM: \(2^2 \times 3^2 \times 5^1\).- First, \(2^2 \times 3^2 = 4 \times 9 = 36\).- Then, multiply by 5: \(36 \times 5 = 180\).

Step 4: Conclusion

The Least Common Multiple (LCM) of 3, 9, and 20 is 180.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its basic building blocks—prime numbers. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
When performing prime factorization:

• Start by dividing the number by the smallest prime number, 2, and continue dividing by 2 until it no longer divides evenly.
• Proceed to the next smallest prime, 3, and repeat the process.
• Continue with 5, 7, or the next primes as necessary.

By the end of this process, you will have expressed the number as a product of prime numbers raised to their respective powers, as seen with the numbers 3, 9, and 20 in the exercise.

Highest Powers

When finding the least common multiple using prime factorization, it is crucial to determine the highest power of each prime number found in the factorization of the numbers. This is because the LCM needs to account for the full scale or breadth that each prime number contributes to any of the given numbers.
To determine the highest power:

• Look at all the prime factorizations.
• For each prime number, identify the greatest exponent that appears.
• For example, the highest power of 2 appears as \(2^2\), the highest power of 3 appears as \(3^2\), and the highest power of 5 appears as \(5^1\).

This step ensures that all prime factors of the original numbers contribute their maximal possible value to the LCM.

Multiplication of Prime Factors

Once you've identified the highest power of each prime, finding the least common multiple is a straightforward multiplication process. By combining these factors, you ensure that the LCM reflects all the contributions each prime factor can make to any of the numbers.
The multiplication process is as follows:

• Multiply the highest power of each prime together: \(2^2 \times 3^2 \times 5^1\).
• Calculate one step at a time: \(2^2 = 4\) and \(3^2 = 9\), then \(4 \times 9 = 36\).
• Finally, multiply by 5 to get the LCM: \(36 \times 5 = 180\).

This process ensures that the resulting product is divisible by each of the original numbers—thus being their least common multiple.

Number Theory

In number theory, the study of integers and their relationships, the least common multiple (LCM) is an important concept. It pertains to finding the smallest integer that is divisible by each number in a set. Understanding LCM is essential for solving problems related to synchronization, scheduling, and allocation.
This concept involves various techniques, with prime factorization being one of the most systematic and reliable methods. By taking the LCM:

• You ensure all input numbers share a common multiple.
• This helps solve simultaneous equations or find periods of repeating events across different cycles.

Number theory provides the foundational principles needed to understand why these techniques work and how they extend to more complex mathematical problems.