Problem 27 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of terms. $$ 15 x, 25 $$

Step-by-Step Solution

Answer

The GCF of 15x and 25 is 5.

Step 1: Find Prime Factors

Identify the prime factors of each number in the list. The number 15 can be broken down into prime factors as \(15 = 3 imes 5\). The number 25 can be broken down into prime factors as \(25 = 5 imes 5\).

Step 2: Identify Common Prime Factors

Look for prime factors that appear in both factorizations. The common prime factor between 15 and 25 is \(5\).

Step 3: Determine the GCF

The greatest common factor, or GCF, is the product of all the common prime factors. Since the only common factor is \(5\), the GCF is \(5\).

Key Concepts

Prime Factorization

Prime factorization is about breaking down a number into its simplest building blocks. These are numbers that are prime, meaning they have no divisors other than 1 and themselves. For example, the number 15 can be expressed as the product of the primes 3 and 5, thus 15 = 3 × 5. Similarly, 25 is 5 × 5. This process helps us reveal the "ingredients" that make up each number.

• Prime numbers are like the building blocks of all numbers.
• To achieve prime factorization, divide the number by the smallest prime until you cannot anymore.
• This technique is crucial for finding the Greatest Common Factor.

Common Factors

Common factors are factors that two or more numbers share. When we find common factors, we're essentially finding what numbers can "evenly" divide each of the given numbers without a remainder. Finding common factors is fundamental when calculating the Greatest Common Factor (GCF).

When identifying common factors:

• List the factors for each number.
• Compare the lists to determine which factors they have in common.
• Only choose prime factors that appear in each list for accuracy.

For the numbers 15 and 25, the prime factorization helps us quickly see that 5 is the only common prime factor.

Step by Step Solution

To effectively determine the GCF, a structured approach is necessary. Let's walk through the basic steps with our example of 15 and 25:

1. **Prime Factorization of Each Number**: First, we break down each number into its prime factors. 15 becomes 3 × 5, and 25 becomes 5 × 5. Now, we have a clear picture of what each number consists of. 2. **Identify the Common Prime Factors**: With the factorizations in front of us, compare them to see which numbers they share. Both numbers share a 5, which is our candidate for the GCF. 3. **Calculate the GCF**: The GCF is found by multiplying the common factors. Since our only common factor is 5, the GCF is just 5.

This method is not just for these numbers but can be applied to any set, ensuring you find the greatest common factor accurately. Remember, using a methodical approach like this ensures clarity and reduces errors.