Problem 74 - Fundamentals of Mathematics

Question

Find the greatest common factor of each collection of numbers. \(64,72,\) and 108

Step-by-Step Solution

Answer

The greatest common factor is 4.

Step 1: Find the Prime Factors of Each Number

To find the greatest common factor (GCF), first determine the prime factors of each number. - For 64: Since 64 is a power of 2, the prime factorization is \(64 = 2^6\).- For 72: Start by dividing by 2. \(72 = 2^3 \times 3^2\).- For 108: Divide by 2 first, then by 3. \(108 = 2^2 \times 3^3\).

Step 2: Identify Common Prime Factors

After determining the prime factors, identify which ones are common to all numbers. - All numbers have the prime factor 2 in common. - Only 108 and 72 have the factor 3 but not 64.

Step 3: Determine the Lowest Power of Common Prime Factor

Since 2 is the only common factor, look at the lowest power of 2 that appears in each factorization:- 64 has \(2^6\)- 72 has \(2^3\)- 108 has \(2^2\)Thus, the lowest power is \(2^2\).

Step 4: Compute the Greatest Common Factor

The GCF is the product of all common prime factors raised to their lowest powers.Since the only common factor with the lowest power is \(2^2\), the GCF is:\[2^2 = 4\]

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are the prime numbers. Prime numbers are those greater than 1 that have no divisors other than 1 and themselves. By expressing a number as a product of prime factors, we can understand its composition better.

For example, in this exercise where we want to find the greatest common factor (GCF) of 64, 72, and 108, the first step is to perform the prime factorization of each number:

• 64: Since 64 is a power of 2, it can be factored as \(64 = 2^6\).
• 72: Start by removing factors of 2; then continue with 3 to get \(72 = 2^3 \times 3^2\).
• 108: First divide by 2, then by 3, resulting in \(108 = 2^2 \times 3^3\).

Understanding how to do prime factorization is essential because it lays the groundwork for identifying common factors, which is crucial in finding the GCF.

Common Prime Factors

Once we have expressed each of the numbers as a product of primes, the next step is to identify which prime factors are shared among the numbers. These are known as common prime factors.

In our exercise, we need to find the common prime factors of 64, 72, and 108. From the prime factorizations:

• 64: \(2^6\)
• 72: \(2^3 \times 3^2\)
• 108: \(2^2 \times 3^3\)

We can see that all three numbers share the factor of 2. However, only 72 and 108 share the factor of 3, which means 3 is not common to all three numbers in this scenario. Finding common prime factors is an important task because it helps us identify what is shared across all numbers, which leads us to the greatest common factor.

Lowest Power of Factors

After identifying the common prime factors, the next step is to determine the lowest power of these shared factors. This is essential when calculating the greatest common factor (GCF).

From our prime factorizations, we saw the common factor was 2 across all three numbers:

• 64: \(2^6\)
• 72: \(2^3\)
• 108: \(2^2\)

Here, the goal is to choose the smallest exponent of 2 that appears in all factorizations. This turns out to be \(2^2\), as it is the smallest among \(2^6\), \(2^3\), and \(2^2\).

The concept of the lowest power is critical. By using the smallest shared power, we ensure the greatest common factor is indeed the largest number that can divide all original numbers without leaving a remainder. Therefore, for this exercise, the GCF is \(2^2 = 4\).