Problem 5 - Fundamentals of Mathematics

Question

Find the GCF of the following numbers. \(450,600,\) and 540

Step-by-Step Solution

Answer

The GCF of 450, 600, and 540 is 60.

Step 1: Prime Factorization of 450

To find the GCF, we first find the prime factorization of each number. Let's start with 450.450 can be factorized as:450 \div 2 = 225 \ 225 \div 3 = 75 \ 75 \div 3 = 25 \ 25 \div 5 = 5 \ 5 \div 5 = 1Thus, the prime factorization of 450 is \(450 = 2 \times 3^2 \times 5^2\).

Step 2: Prime Factorization of 600

Next, we find the prime factorization of 600.600 can be factorized as:600 \div 2 = 300 \ 300 \div 2 = 150 \ 150 \div 2 = 75 \ 75 \div 3 = 25 \ 25 \div 5 = 5 \ 5 \div 5 = 1Thus, the prime factorization of 600 is \(600 = 2^3 \times 3 \times 5^2\).

Step 3: Prime Factorization of 540

Now, we find the prime factorization of 540.540 can be factorized as:540 \div 2 = 270 \ 270 \div 2 = 135 \ 135 \div 3 = 45 \ 45 \div 3 = 15 \ 15 \div 3 = 5 \ 5 \div 5 = 1Thus, the prime factorization of 540 is \(540 = 2^2 \times 3^3 \times 5\).

Step 4: Identify Common Prime Factors

Identify the common prime factors from each factorization: - All three numbers have the prime factors 2, 3, and 5. Now, we must note the lowest power of each common factor shared by all three numbers.

Step 5: Determine the Lowest Powers

Determine the lowest power for each common prime factor:\(2: \text{min}(2, 3, 2) = 2\) \ \(3: \text{min}(2, 1, 3) = 1\) \ \(5: \text{min}(2, 2, 1) = 1\)

Step 6: Calculate the GCF

The GCF is found by multiplying these lowest powers together:\[\text{GCF} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60\]

Step 7: Conclusion

The greatest common factor (GCF) of 450, 600, and 540 is 60.

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime factors. A prime number is a number greater than 1 that can only be divided evenly by 1 and itself. To find the prime factorization of a number, you repeatedly divide the number by its smallest prime factor until only 1 is left.

• Start with the smallest prime number, which is 2. If the number is even, divide it by 2.
• If the number is odd, try dividing it by the next smallest prime, which is 3, then 5, and so on.
• Continue this process until the remaining quotient is 1.

For example, to find the prime factorization of 450:450 is even, so start by dividing it by 2. The next number, 225, is divisible by 3, followed by 75 and 25, up to 5, where repeat division by 5 yields 1. Thus, the factorization appears as:\[450 = 2 \times 3^2 \times 5^2\]

Common Factors

Common factors are those that appear in the prime factorizations of different numbers. When identifying common factors, you're looking for prime numbers that are found in each number's prime factorization.
For instance, when examining 450, 600, and 540:

• All three numbers include the prime factor 2.
• Each has the factor 3.
• The factor 5 is also shared among them.

These common prime factors are crucial for determining the greatest common factor (GCF), as the GCF is derived from only these shared factors.

Lowest Power

Identifying the lowest power for each common factor is essential when calculating the GCF. The power refers to the exponent in the prime factorization, indicating how many times the prime number is multiplied by itself.
To find the power of each common factor:

• Look at each prime factor shared between the numbers.
• Choose the lowest exponent from those numbers.

For example, in the sets of factors for 450, 600, and 540:

• 2 is raised to the lowest power of 2, or \(2^2\).
• 3 is raised to the lowest power of 1, or \(3^1\).
• 5 is raised to the lowest power of 1, or \(5^1\).

By focusing on these minimum powers, you're ensuring that each factor appears in a way that is common to all the numbers being considered.

Multiply Factors

Once you have determined the lowest power for each common factor, the next step is multiplying these factors to find the greatest common factor (GCF). This step combines all the common prime factors while ensuring they appear in the minimal necessary form.
The calculation involves:

• Taking each common factor raised to its lowest power.
• Multiplying these values together.

Consider 450, 600, and 540. The lowest powers determined earlier are multiplied as follows:\[\text{GCF} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60\]This computed value, 60, represents the largest possible number that can divide all three original numbers without leaving a remainder, thus confirming it as the greatest common factor.