Problem 24 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. \(175,225,\) and 400

Step-by-Step Solution

Answer

The GCF of 175, 225, and 400 is 25.

Step 1: Prime Factorization of 175

Start by finding the prime factorization of 175. Divide 175 by the smallest prime number, which is 5:\[ 175 \div 5 = 35 \]Now, divide 35 by 5:\[ 35 \div 5 = 7 \]Since 7 is a prime number, the factorization of 175 is:\[ 175 = 5^2 \times 7 \]

Step 2: Prime Factorization of 225

Next, find the prime factorization of 225. Divide 225 by 5:\[ 225 \div 5 = 45 \]Divide 45 by 5 again:\[ 45 \div 5 = 9 \]Finally, divide 9 by 3:\[ 9 \div 3 = 3 \]Since 3 is a prime number, the factorization of 225 is:\[ 225 = 5^2 \times 3^2 \]

Step 3: Prime Factorization of 400

Find the prime factorization of 400. Start with 400 divided by 2:\[ 400 \div 2 = 200 \]Continue dividing by 2:\[ 200 \div 2 = 100 \]\[ 100 \div 2 = 50 \]\[ 50 \div 2 = 25 \]Finally, divide by 5:\[ 25 \div 5 = 5 \]\[ 5 \div 5 = 1 \]The factorization of 400 is:\[ 400 = 2^4 \times 5^2 \]

Step 4: Analyze Common Factors

Identify the common factors from the prime factorizations:- 175 = \( 5^2 \times 7 \)- 225 = \( 5^2 \times 3^2 \)- 400 = \( 2^4 \times 5^2 \)The common factor among all three numbers is \( 5^2 \).

Step 5: Determine the GCF

Since the common factor is \( 5^2 \), calculate it:\[ 5^2 = 25 \]Hence, the greatest common factor of 175, 225, and 400 is 25.

Key Concepts

Prime Factorization

Prime factorization is a process used to express a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves, such as 2, 3, 5, 7, 11, and so on. This technique is essential for finding the greatest common factor because it showcases the basic building blocks of any given number.

For example, when we look at 175, we break it down into its prime components by dividing it by the smallest prime number possible until we are left with a prime number at the end. This method gives us the factorization of 175 as:

• First divide 175 by 5 to get 35.
• Next, divide 35 by 5 to get 7.
• The prime factorization is then: \(175 = 5^2 \times 7\).

Reapplying the same approach for 225 and 400 results in their respective prime factorizations. Using prime factorization helps simplify the process of identifying common factors among different numbers, which is crucial in determining the GCF.

Common Factors

Once we have the prime factorization of each number, the next thing to do is identify their common factors. These are prime numbers that appear in the prime factorizations of all the numbers involved. Recognizing common factors is a straightforward but crucial step when finding the greatest common factor.

In our example with numbers 175, 225, and 400, after factorization we obtained:

• \(175 = 5^2 \times 7\)
• \(225 = 5^2 \times 3^2\)
• \(400 = 2^4 \times 5^2\)

The common factor in all three numbers is \(5^2\). This shows that 5 appears to the same power in the factorization of each number, which is vital for calculating the GCF.

Step by Step Solution

Working through problems systematically with a step by step solution can make complex concepts such as finding the greatest common factor more understandable. Breaking down each part ensures that each piece of the puzzle is manageable and clearly laid out.

Using the step by step method to find the GCF:

• Step 1: Break down each number into prime factors.
• Step 2: Identify the common factors across all numbers.
• Step 3: Choose the smallest exponent of the common factors. For this example, \(5^2 = 25\) is the common factor with the lowest power shared by all three numbers.
• Step 4: Calculate the GCF by multiplying these common factors with the smallest exponent. Therefore, the greatest common factor of 175, 225, and 400 is \(25\).

Through this systematic approach, the problem is solved with clarity, allowing learners to make connections between the steps and understand not just the 'how,' but also the 'why' behind each process.