Problem 18 - Elementary Algebra

Question

For the following problems, find the least common multiple of given numbers. 4, 5, 21

Step-by-Step Solution

Answer

Answer: The least common multiple of 4, 5, and 21 is 420.

Step 1: Prime Factorization of the Given Numbers

Prime factorize the given numbers using a factor tree or division method. 4 = 2^2 5 = 5^1 21 = 3^1 * 7^1

Step 2: Identify the Maximum Exponents for Each Prime Factor

Find the maximum exponent for each prime factor among the given numbers. For 2: the maximum exponent is 2 For 3: the maximum exponent is 1 For 5: the maximum exponent is 1 For 7: the maximum exponent is 1

Step 3: Calculate the Least Common Multiple (LCM)

Multiply each prime factor raised to its maximum exponent calculated in the previous step to get the LCM. LCM = 2^2 * 3^1 * 5^1 * 7^1 LCM = 4 * 3 * 5 * 7 LCM = 420 The least common multiple of 4, 5, and 21 is 420.

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime numbers. A prime number is a number that has only two divisors: 1 and itself. For example, when we break down the number 4 using prime factorization, we represent it as \(2^2\). This means 4 is the product of two 2's multiplied together. Similarly:

• 5 is already a prime number, so its factorization is \(5^1\).
• 21 can be broken down into \(3^1 \times 7^1\), both of which are prime numbers.

Prime factorization provides a way to understand the building blocks of any given number, which is crucial for finding the least common multiple.

Factor Tree Method

The factor tree method is a visual way to perform prime factorization. It's like breaking down a number into branches until you reach the prime factors. Here's how it works:

• Start with the number you want to factorize, like 21.
• Draw two branches and write down any pair of factors, such as 3 and 7, which multiply to give 21.
• Continue breaking down the factors until all branches end with prime numbers.

This method helps you visually see how numbers are composed and ensures that you don't miss any prime factors along the way. It's especially useful for simplifying the process for large numbers.

Exponent

In prime factorization, an exponent represents the number of times a prime number is multiplied by itself. For instance, in \(2^2\), 2 is the base (the prime number), and 2 is the exponent showing it is multiplied by itself:

• \(2 \times 2 = 4\)

Using exponents simplifies representing repeated multiplication. It also plays a critical role when determining the least common multiple because you need to choose the highest exponent of each prime factor across all numbers involved.

Multiplication of Prime Factors

Once you've identified the prime factors and their highest exponents, you can calculate the least common multiple by multiplying these factors. For the numbers 4, 5, and 21:

• The factors are \(2^2\), \(3^1\), \(5^1\), and \(7^1\).
• Multiply these together: \(2^2 \times 3^1 \times 5^1 \times 7^1\).
• This results in 4 x 3 x 5 x 7 = 420.

This product is the least common multiple, combining all necessary prime factors at their maximum required powers to cover all original numbers.