Problem 72 - Fundamentals of Mathematics

Question

Find the greatest common factor of each collection of numbers. 18,48 , and 72

Step-by-Step Solution

Answer

The greatest common factor is 6.

Step 1: Prime Factorization

Start by determining the prime factors of each number. - For 18: Divide by 2 to get 9, then divide by 3 twice to get 1. Thus, the prime factorization of 18 is \(2 \times 3^2\).- For 48: Divide by 2 four times to get 3, making the prime factorization \(2^4 \times 3\).- For 72: Divide by 2 three times to get 9, then divide by 3 twice to reach 1. Hence, the prime factorization is \(2^3 \times 3^2\).

Step 2: Identify Common Factors

Identify the common prime factors in all three numbers. The prime factors we've found are:- 18: \(2 \times 3^2\)- 48: \(2^4 \times 3\)- 72: \(2^3 \times 3^2\)The common factors among these are 2 and 3.

Step 3: Calculate GCF

To find the greatest common factor, select the lowest power of all common prime factors:- For the base 2: The lowest power is \(2^1\).- For the base 3: The lowest power is \(3^1\).Therefore, the greatest common factor is \(2^1 \times 3^1 = 6\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves.
To find the prime factorization, divide the number successively by the smallest prime numbers (2, 3, 5,...) until the result is a quotient of 1. For example, with the number 18:

• Divide 18 by 2 to get 9.
• Divide 9 by 3 twice to reach 1.

This means the prime factorization of 18 is represented as \(2 \times 3^2\).
Prime factorization helps us understand what prime numbers make up a given whole number, which is crucial for solving problems involving the greatest common factor (GCF).

Common Factors

Common factors are the prime factors that two or more numbers share. To find common factors, look at each number’s prime factorization and identify the factors they have in common.
Consider the numbers 18, 48, and 72:

• The prime factorization of 18 is \(2 \times 3^2\).
• The prime factorization of 48 is \(2^4 \times 3\).
• The prime factorization of 72 is \(2^3 \times 3^2\).

In these factorizations, both 2 and 3 appear in all three numbers. Identifying these common factors is a key step in finding the greatest common factor (GCF). Finding common prime factors among multiple numbers simplifies and illustrates their shared traits, helping us solve for the GCF effectively.

Calculating GCF

Calculating the greatest common factor (GCF) involves selecting the smallest powers of the common prime factors found in each number's prime factorization. This ensures you find the highest number that divides each original number exactly.
For the example of 18, 48, and 72:

• The smallest power of the common base 2 is \(2^1\).
• The smallest power of the common base 3 is \(3^1\).

So, the GCF is the product of these smallest powers: \(2^1 \times 3^1 = 6\).
Calculating the GCF allows you to understand the largest scale at which multiple numbers can be evenly divided, assisting in problem-solving tasks like simplifying fractions or solving equations involving multiple terms.