Problem 70 - Fundamentals of Mathematics

Question

Find the greatest common factor of each collection of numbers. 42 and 54

Step-by-Step Solution

Answer

The greatest common factor of 42 and 54 is 6.

Step 1: Prime Factorization of 42

To find the greatest common factor (GCF), start by determining the prime factors of each number. For 42, we divide by the smallest prime number which is 2: \(42 \div 2 = 21\). Then, divide 21 by 3: \(21 \div 3 = 7\). 7 is a prime number, so we stop here. Therefore, the prime factorization of 42 is \(2 \times 3 \times 7\).

Step 2: Prime Factorization of 54

Now find the prime factors of 54. Start with the smallest prime, 2: \(54 \div 2 = 27\). Next, divide 27 by 3: \(27 \div 3 = 9\), and again by 3: \(9 \div 3 = 3\). Finally, 3 is a prime number, so we stop. The prime factorization of 54 is \(2 \times 3^3\) or equivalently \(2 \times 3 \times 3 \times 3\).

Step 3: Identify Common Prime Factors

List the prime factors of both numbers. 42 has the factors \(2 \times 3 \times 7\), and 54 has \(2 \times 3 \times 3 \times 3\). The common factors between them are 2 and 3.

Step 4: Calculate the GCF

Multiply the common prime factors to find the GCF. The common factors are 2 and 3, thus the greatest common factor is \(2 \times 3 = 6\).

Key Concepts

Prime Factorization

Prime factorization is a process by which we express a number as a product of its prime numbers. **Prime numbers** are those greater than 1 that have no divisors other than 1 and themselves. Prime factorization is fundamental in finding the greatest common factor (GCF). For example, let's break down the number 42:

• 42 is divided by the smallest prime number, 2: \(42 \div 2 = 21\)
• Next, 21 is divided by 3: \(21 \div 3 = 7\), where 7 is already a prime number

Thus, the complete prime factorization of 42 is \(2 \times 3 \times 7\). Similarly, for 54, starting with 2 and then by 3, its prime factorization becomes \(2 \times 3 \times 3 \times 3\). Understanding this breakdown is key to comparing numbers or finding GCFs.

Common Prime Factors

Once the prime factors of each number are identified, the next step is to determine which factors are present in both sets. These are known as **common prime factors**.

• For 42, the prime factors are \(2\), \(3\), and \(7\).
• For 54, the prime factors are \(2\), \(3\), \(3\), and another \(3\).

Comparing both sets, the common prime factors between 42 and 54 are 2 and 3. Recognizing these commonalities allows us to proceed to find the greatest common factor by multiplication.

Numerical Problem Solving

Numerical problem solving involves using mathematical methods to find solutions, such as determining the GCF. By examining prime factors, you can find solutions efficiently. Let's reflect on our example:

• Identify the prime factors of each number separately.
• Find common factors between the numbers.
• Multiply the common factors to get the GCF.

In our specific case, the common prime factors 2 and 3, when multiplied, give the greatest common factor of 6. Steps like these simplify complex numerical tasks, making problem-solving more straightforward and approachable.