Problem 21 - Fundamentals of Mathematics

Question

Find the greatest common factor. 200 and 36

Step-by-Step Solution

Answer

The greatest common factor of 200 and 36 is 4.

Step 1: Prime Factorization of 200

First, we need to find the prime factorization of 200. We start by dividing 200 by the smallest prime number, which is 2: \[ 200 \div 2 = 100 \] 100 is still even, so divide by 2 again: \[ 100 \div 2 = 50 \] 50 is also even, so divide by 2 once more: \[ 50 \div 2 = 25 \] 25 is not divisible by 2, so try the next smallest prime, 5: \[ 25 \div 5 = 5 \] Finally, 5 divided by 5 is 1: \[ 5 \div 5 = 1 \] So, the prime factorization of 200 is \( 2^3 \times 5^2 \).

Step 2: Prime Factorization of 36

Next, we find the prime factorization of 36. Start by dividing by the smallest prime number, 2: \[ 36 \div 2 = 18 \] 18 is still even, so divide by 2 again: \[ 18 \div 2 = 9 \] 9 is not divisible by 2, so we try 3: \[ 9 \div 3 = 3 \] Finally, 3 divided by 3 is 1: \[ 3 \div 3 = 1 \] The prime factorization of 36 is \( 2^2 \times 3^2 \).

Step 3: Identify Common Factors

Now, identify the common prime factors in both factorizations. For 200, the factors are \( 2^3 \times 5^2 \) and for 36, the factors are \( 2^2 \times 3^2 \). The common factor is 2. We take the smallest power of this common factor. 200 has \( 2^3 \) and 36 has \( 2^2 \), so the smallest power is \( 2^2 \).

Step 4: Calculate the Greatest Common Factor

The greatest common factor (GCF) is the product of the common factors raised to their smallest power. Here, we found the common factor to be 2. The smallest power among the common factors is \( 2^2 \), so the GCF is:\[ 2^2 = 4 \]

Key Concepts

Prime Factorization

Prime factorization is a fundamental concept in mathematics that breaks down any composite number into a product of its prime numbers. This process highlights how every integer is uniquely constructed from prime factors. By understanding the smallest building blocks of a number, you gain insight into its properties.
For example, with the number 200, the prime factorization process involves dividing by the smallest prime, 2, repeatedly until it’s no longer feasible, then moving to the next smallest prime, 5. This results in:

• 200 = \( 2^3 \times 5^2 \)

Similarly with 36, division by 2 and then by 3 results in:

• 36 = \( 2^2 \times 3^2 \)

Notice the simplicity when numbers are distilled to these basic components. Understanding prime factorization is key to solving numerous mathematical problems involving divisibility, such as finding the greatest common factor.

Common Factors

Common factors are numbers that divide two or more numbers evenly, meaning there’s no remainder. In our example, the challenge is to identify these in the prime factorization of 200 and 36. Analyzing the prime factors:

• For 200, the prime factors are \( 2^3 \) and \( 5^2 \)
• For 36, the prime factors are \( 2^2 \) and \( 3^2 \)

The common prime factor between these is 2. To find their greatest common factor, take the lowest power of each common factor, ensuring the division remains exact. Thus, among \( 2^3 \) and \( 2^2 \), the smallest power is \( 2^2 \). Therefore, the greatest common factor (GCF) computes to \( 4 \).
Identifying common factors facilitates understanding the shared properties and relationships between numbers.

Mathematics Education

Learning mathematics is not just about memorizing procedures but involves grasping foundational concepts and their applications. Prime factorization and finding common factors are crucial topics that offer a deeper comprehension of number theory.

Engaging with exercises like determining the greatest common factor strengthens problem-solving skills and enriches numerical literacy, which are invaluable skills at every education level. Exploring concepts through examples, like the prime factors of 200 and 36, helps demystify abstract math problems into tangible, comprehensible steps.

• Concepts like prime factorization help in understanding more sophisticated math topics.
• They allow students to connect lessons to real-world applications, fostering a more profound educational experience.

Encouraging an instructionally rich environment helps students develop logical thinking and a detailed appreciation of mathematics.