Problem 21 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of numbers. $$ 14,21,42 $$

Step-by-Step Solution

Answer

The GCF of 14, 21, and 42 is 7.

Step 1: Break Down Each Number Into Prime Factors

To find the GCF, start by breaking each number into its prime factors. The prime factorization of 14 is: \(14 = 2 imes 7\). The prime factorization of 21 is: \(21 = 3 imes 7\). Finally, the prime factorization of 42 is: \(42 = 2 imes 3 imes 7\).

Step 2: Identify Common Factors

Next, identify the common factors in the prime factorizations of each number. All three numbers (14, 21, and 42) include the factor \(7\) as part of their prime factorization.

Step 3: Determine the Greatest Common Factor (GCF)

The greatest common factor is the highest number that appears in the prime factorizations of all three numbers. The only factor that fits this criterion is \(7\). Thus, the GCF of 14, 21, and 42 is \(7\).

Key Concepts

Prime Factorization

Prime factorization refers to expressing a number as the product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For instance, the prime numbers include 2, 3, 5, 7, 11, and so on. To factorize a number means to break it down into its prime building blocks.

Prime factorization is essential for finding the Greatest Common Factor (GCF). Why? Because the GCF can be determined by identifying the highest common prime factors between numbers. When you break down 14, 21, and 42 into primes:

• 14 becomes 2 and 7.
• 21 becomes 3 and 7.
• 42 becomes 2, 3, and 7.

In each case, prime factorization simplifies the numbers to their smallest, indivisible components, facilitating the comparison of factors across multiple numbers. Always start with the smallest prime number when breaking a number down and move to larger primes as needed.

Common Factors

Common factors are numbers that divide each member of a given set of numbers without leaving a remainder. Identifying common factors is a key step in finding the GCF. When you look for common factors, you're essentially trying to find those numbers that are shared in the prime factorization of each number in your list.

In the example of 14, 21, and 42, after prime factorization, notice how 7 appears in each list of factors:

• Both 14 and 42 have the factor 2.
• Both 21 and 42 have the factor 3.
• All three, 14, 21, and 42, share the factor 7.

While 2 and 3 may occur in pairs, 7 is the only factor common to all three. Finding these connections helps in narrowing down the GCF. This process ensures that we do not overlook smaller common factors that may also play a significant role.

Number Theory

Number theory is a field of mathematics dedicated to studying numbers, particularly integers, and their properties. It is the backbone behind understanding concepts like the GCF. What makes number theory special is its focus on the simple yet profound properties of numbers, including divisibility, primes, and common factors.

Concepts from number theory, such as prime factorization and finding common factors, are practical tools used to solve everyday mathematical problems like calculating the GCF. In our example, number theory helps us systematically decompose each number and identify shared elements, which is crucial for simplifying problems and solutions regarding divisibility.

• It explains why prime factorization is effective for finding common factors.
• It provides logic for why the GCF is the largest shared factor.

Delving into number theory gives us a clearer picture of the relationships between numbers and the logic behind calculations, ensuring our understanding goes beyond memorization of rules.