Problem 1 - Prealgebra

Question

Find the least common multiple (LCM) of each pair of numbers. $$6,8$$

Step-by-Step Solution

Answer

The LCM of 6 and 8 is 24.

Step 1: Find the prime factors

Identify the prime factors of each number. For 6, the prime factors are 2 and 3, so we can write 6 as \(2 \times 3\). For 8, the prime factors are all 2s, so we can write 8 as \(2^3\).

Step 2: List the highest powers of all prime factors

Create a list containing the highest power of each prime number that appears in the factorization. The prime numbers here are 2 and 3. For 2, the highest power is \(2^3\) (from 8), and for 3, it is \(3^1\) (from 6).

Step 3: Multiply these highest powers

Calculate the product of these highest powers: \(2^3 \times 3^1\). First, calculate \(2^3 = 8\) and \(3^1 = 3\). Then multiply these results: \(8 \times 3 = 24\).

Step 4: Finalize the LCM

The least common multiple of the numbers 6 and 8 is the result of the calculation above, which is 24.

Key Concepts

Prime Factorization

Prime factorization involves breaking down a number into its basic building blocks — the prime numbers. Every number can be expressed as a product of prime numbers. Understanding prime factorization is crucial when you want to find the least common multiple (LCM) or the greatest common divisor (GCD).

Prime numbers are those that are only divisible by 1 and themselves. For example:

• 2, 3, 5, 7, and 11 are prime numbers.
• 6 is not a prime number because it can be divided evenly by 2 and 3.

To find the prime factorization of a number, divide it by prime numbers until you reach 1.
For example:

• The prime factorization of 6 is: 2 and 3 (because 6 = 2 × 3).
• The prime factorization of 8 is: 2, 2, and 2, or 2³ (because 8 = 2 × 2 × 2).

Prime factorization is a key step in finding the LCM, as it helps identify the highest powers of prime numbers required in the multiplication step.

Multiples

Multiples of a number are what you get when you multiply that number by integers. Understanding multiples is essential when dealing with concepts like LCM, which helps in finding a common ground for numbers.

For instance:

• The first few multiples of 6 are: 6, 12, 18, 24, 30, etc. (because they are 6 multiplied by 1, 2, 3, 4, 5, ...).
• The first few multiples of 8 are: 8, 16, 24, 32, etc. (because they are 8 multiplied by 1, 2, 3, 4, ...).

By comparing multiples of different numbers, you can find that they share some multiples. The smallest of these shared multiples is called the Least Common Multiple (LCM).
For example, for numbers 6 and 8, the LCM is 24, because 24 is the first number found in both lists of multiples. Identifying multiples helps provide a clear path to determine the least or greatest shared value of two numbers.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that can divide two or more numbers without leaving a remainder.
This concept is especially useful when looking for ways to simplify expressions or find commonalities. To find the GCD, list all the divisors of the given numbers or use their prime factorizations and choose the smallest powers of only the prime factors they have in common. Example:

• For 6, the divisors are: 1, 2, 3, and 6.
• For 8, the divisors are: 1, 2, 4, and 8.
• The common divisors of 6 and 8 are: 1 and 2.

Thus, the GCD of 6 and 8 is 2.

Understanding the GCD helps when trying to reduce fractions to their simplest form, or when prepping numbers for operations that require equal sharing.