Problem 10 - Elementary Algebra

Question

For the following problems, find the least common multiple of given numbers. 20,24

Step-by-Step Solution

Answer

Answer: The least common multiple (LCM) of 20 and 24 is 120.

Step 1: Prime factorization of each number

Find the prime factors of each number by breaking it down into the product of its prime factors. For 20: \(20 = 2^2 \cdot 5^1\) For 24: \(24 = 2^3 \cdot 3^1\)

Step 2: Identify the unique prime factors and their highest powers

Look at the prime factorization of each number and find the unique prime factors that appear in either list. Then, for each unique prime factor, find the highest power it's raised to in either number. Unique prime factors: 2, 3, and 5 Highest powers: - For 2: \(2^3\) (from 24) - For 3: \(3^1\) (from 24) - For 5: \(5^1\) (from 20)

Step 3: Multiply the highest powers of the unique prime factors

Multiply the highest powers of each unique prime factor to find the least common multiple (LCM) of the two numbers. LCM(20, 24) = \(2^3 \cdot 3^1 \cdot 5^1 = 8 \cdot 3 \cdot 5 = 120\) The least common multiple of 20 and 24 is 120.

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime numbers. These are numbers that are only divisible by 1 and themselves. For example, let's factorize 20 and 24.

• For 20, we divide by the smallest prime number, 2: - 20 ÷ 2 = 10 - 10 ÷ 2 = 5 - 5 is a prime number, so we stop here - The prime factorization of 20 is \(2^2 \cdot 5^1\)
• For 24, following the same method: - 24 ÷ 2 = 12 - 12 ÷ 2 = 6 - 6 ÷ 2 = 3 - 3 is a prime number - The prime factorization of 24 is \(2^3 \cdot 3^1\)

This breakdown into prime numbers helps in finding the least common multiple (LCM) of any two numbers.

Unique Prime Factors

Unique prime factors are the distinct prime numbers found in the factorization of one or more numbers. Finding these helps when calculating the LCM. Let's gather the unique prime factors from 20 and 24:

• From the factorization of 20: 2 and 5
• From the factorization of 24: 2 and 3

Notice that the number 2 appears in both sets, while 3 and 5 appear individually.
The unique prime factors from both numbers combined are 2, 3, and 5. They represent every prime factor occurring across the numbers being considered. These factors will be used to determine the LCM.

Highest Power of Prime Factors

Once we identify the unique prime factors, the next step is to find their highest power in either number. This ensures that both numbers are considered fully in the LCM calculation. Here's how:

• Prime factor 2: Appears as \(2^2\) in 20 and \(2^3\) in 24. The highest power is \(2^3\).
• Prime factor 3: Appears only in 24 as \(3^1\). The highest power is \(3^1\).
• Prime factor 5: Appears only in 20 as \(5^1\). The highest power is \(5^1\).

These highest powers, \(2^3\), \(3^1\), and \(5^1\), are multiplied together to find the LCM: \[2^3 \cdot 3^1 \cdot 5^1 = 8 \cdot 3 \cdot 5 = 120\] This product gives us the least common multiple of the two numbers, ensuring that all unique factors are included with the largest exponents necessary.