Problem 65 - Fundamentals of Mathematics

Question

Find the greatest common factor of 48 and 72 .

Step-by-Step Solution

Answer

The greatest common factor of 48 and 72 is 24.

Step 1: Understand the Problem

We need to find the greatest common factor, or GCF, of two numbers: 48 and 72. The GCF is the largest number that evenly divides both of these numbers.

Step 2: Prime Factorize the Numbers

We will find the prime factorization of both numbers. For 48, the factorization is obtained by dividing by the smallest prime numbers: - 48 ÷ 2 = 24 - 24 ÷ 2 = 12 - 12 ÷ 2 = 6 - 6 ÷ 2 = 3 - 3 ÷ 3 = 1 So, the prime factorization of 48 is 2^4 × 3.

Step 3: Prime Factorize the Second Number

Next, factorize 72 using the same method: - 72 ÷ 2 = 36 - 36 ÷ 2 = 18 - 18 ÷ 2 = 9 - 9 ÷ 3 = 3 - 3 ÷ 3 = 1 So, the prime factorization of 72 is 2^3 × 3^2.

Step 4: Identify Common Factors

Identify the common prime factors from both factorizations. For 48 and 72, the common primes are 2 and 3.

Step 5: Determine the Greatest Common Factor

For the common primes, take the lowest power from each: from 2^4 and 2^3, take 2^3, and from 3^1 and 3^2, take 3^1. Multiply these together to get the GCF:\[GCF = 2^3 \times 3^1 = 8 \times 3 = 24\]

Key Concepts

Understanding Prime Factorization

Prime factorization is a method used to break down a number into its prime number components. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The prime factorization process helps us identify the building blocks of a number. This is crucial when solving problems such as finding the greatest common factor (GCF). Understanding this concept can make problem-solving more straightforward, as it reveals the structure of numbers. To perform prime factorization:

• Start by dividing the number by the smallest prime number, typically 2.
• Continue dividing by prime numbers until you can no longer divide evenly.
• The result is a set of primes multiplied together, representing the original number.

In the case of 48, prime factorization involves dividing repeatedly by 2 and finally by 3, giving us the product of primes: 2^4 × 3.

Identifying Common Factors in Numbers

Common factors are numbers that are divisors of two or more numbers. When examining the prime factorizations of two numbers, common factors can be easily identified by comparing their sets of prime factors. For example, with numbers 48 and 72, their prime factorizations are:

• 48: 2^4 × 3
• 72: 2^3 × 3^2

The common prime factors here are 2 and 3. Determining these factors is essential for finding the GCF, as they are the bases that these numbers share.

Mathematics Problem Solving: Finding the GCF

Mathematics problem solving often requires understanding and applying multiple concepts, such as prime factorization and identifying common factors, to find solutions like the greatest common factor. The GCF is the largest number that can divide two numbers without leaving a remainder. To find the GCF using prime factorization:

• Identify the prime factors of each number.
• Find common factors by comparing these lists.
• Select the lowest power of each common factor.
• Multiply these selected powers together to get the GCF.

In our exercise with 48 and 72, the common prime factorization is 2^3 × 3^1. Thus, the GCF is calculated as 8 × 3, resulting in 24, the greatest common factor of these two numbers.