Problem 17 - Elementary Algebra

Question

For the following problems, find the least common multiple of given numbers. 8, 10, 15

Step-by-Step Solution

Answer

Answer: The least common multiple of 8, 10, and 15 is 120.

Step 1: Prime factorization of each number

First, we need to find the prime factorization of each number. Prime factorization of 8: 2^3 Prime factorization of 10: 2^1 × 5^1 Prime factorization of 15: 3^1 × 5^1

Step 2: Identify the highest power of each factor

In this step, we'll identify the highest power of each factor in all prime factorizations: Highest power of 2: 2^3 (from 8) Highest power of 3: 3^1 (from 15) Highest power of 5: 5^1 (from 10)

Step 3: Multiply the highest powers together

Finally, multiply the highest powers together to find the LCM: LCM = 2^3 × 3^1 × 5^1 = 8 × 3 × 5 = 120 The least common multiple of 8, 10, and 15 is 120.

Key Concepts

Prime Factorization

Prime factorization is a method of breaking down a number into its basic building blocks by expressing it as a product of its prime numbers. Primes are numbers greater than 1 that have no divisors other than 1 and themselves.
To perform prime factorization, you start by dividing the number by the smallest prime number until you can no longer divide evenly, then move to the next smallest prime number.

• First, check if the number is divisible by 2 (the smallest prime). If so, divide it and continue dividing by 2 until it's no longer possible.
• Next, move to the next smallest prime, 3, and repeat the process.
• Continue with primes like 5, 7, etc., until the number is completely factored into primes.

For example, the number 8 can be broken down as 2 × 2 × 2, or in exponential form as \(2^3\). This makes it clear that 8's prime factors are entirely based on the prime 2. Similarly, 10 becomes \(2^1 \times 5^1\), and 15 becomes \(3^1 \times 5^1\). Prime factorization simplifies finding the least common multiple for several numbers, as it highlights the exact ratio in which these primes contribute to the original numbers.

Greatest Power

Finding the greatest power of each prime factor is crucial when calculating the least common multiple (LCM). This step ensures that the LCM covers all the necessary multiples as found in the given numbers.

After performing the prime factorization for each number, the task is to identify the highest exponent for each prime number present.

• Look at the factor tree of each number and identify how many times each prime number appears.
• Select the greatest power of each prime across the factorizations.

For the numbers 8, 10, and 15, let's identify the greatest power:

• For prime number 2, the greatest power is \(2^3\) from 8.
• For the prime number 3, the greatest power is \(3^1\) from 15.
• For the prime number 5, the greatest power is \(5^1\) shared by both 10 and 15.

Remember, the LCM needs to incorporate the most dominant representation of each prime to ensure divisibility by each of the original numbers. This is why taking the highest power is essential.

Multiplication of Primes

Once you have determined the greatest power of each prime number involved, the final step is to multiply them together to find the least common multiple (LCM). This stage combines the components into one singular value that reflects all the individual factors from the original numbers.

The multiplication of primes, whose greatest powers were identified, consolidates the requirement of having enough of each prime to divide all original numbers smoothly.

• Start with each prime number's greatest power.
• Multiply them altogether.

In our example:

• We have \(2^3\), which equals 8.
• Then there's \(3^1\), which is 3.
• Finally, \(5^1\), which is 5.

Hence, the least common multiple is computed as:\[\text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120.\]This number, 120, is the smallest number that all three original numbers (8, 10, and 15) can divide without leaving a remainder, making it the ideal LCM. This calculation shows how the multiplication of greatest power primes helps find a shared multiple for all numbers in consideration.