Problem 10 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. 20 and 24

Step-by-Step Solution

Answer

The GCF of 20 and 24 is 4.

Step 1: Prime Factorization of 20

First, let's find the prime factorization of 20. Start by dividing 20 by the smallest prime number, which is 2: \[ 20 \div 2 = 10 \]20 can be divided by 2, giving 10. Then, divide 10 by 2:\[ 10 \div 2 = 5 \]Now, 5 is a prime number, so the prime factorization of 20 is:\[ 20 = 2^2 \times 5 \]

Step 2: Prime Factorization of 24

Next, find the prime factorization of 24. Begin by dividing 24 by 2:\[ 24 \div 2 = 12 \]Then, divide 12 by 2:\[ 12 \div 2 = 6 \]Divide 6 by 2 again:\[ 6 \div 2 = 3 \]3 is a prime number, so the prime factorization of 24 is:\[ 24 = 2^3 \times 3 \]

Step 3: Identify Common Prime Factors

Identify the common prime factors from the factorizations:- For 20: \( 2^2 \times 5 \)- For 24: \( 2^3 \times 3 \)The common prime factor between 20 and 24 is 2.

Step 4: Find the Least Power of Common Factors

Since the only common prime factor is 2, we need to find the least power of this factor in both numbers. - In 20: the factor 2 appears as \( 2^2 \).- In 24: the factor 2 appears as \( 2^3 \).The least power is \( 2^2 \).

Step 5: Calculate the GCF

The greatest common factor (GCF) is found by using the least power of the common factors. Here, since the least power of 2 is 2 squared:\[ GCF = 2^2 = 4 \]

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. Prime numbers are those greater than 1, having no divisors other than 1 and themselves. To find the prime factorization, you begin by dividing the number by the smallest prime number, which is 2, and continue dividing the quotient by prime numbers until only prime numbers are left.

• For example, with the number 20, you start dividing by 2: 20 divided by 2 is 10. Keep dividing by 2 until you can't, which gives 10 divided by 2 equals 5.
• Once you reach a prime number like 5, you stop, and the prime factorization of 20 becomes \( 2^2 \times 5 \).

Prime factorization is a critical step in finding the greatest common factor and understanding the structure of numbers.

Common Prime Factors

Common prime factors are the prime numbers that appear in the prime factorization of each number in question. After completing the prime factorization for multiple numbers, you identify which prime factors these numbers share.

• For instance, with the numbers 20 and 24, their prime factorizations are \( 2^2 \times 5 \) and \( 2^3 \times 3 \) respectively.


• By comparing these, you can see that the common prime factor between 20 and 24 is 2. This commonality is what helps us further determine the GCF.

Understanding common prime factors is essential because it helps narrow down the possibilities when calculating the greatest common factor.

Least Power of Common Factors

Once you have identified the common prime factors, the next step is finding the least power of these factors. This means looking at the common prime factors and choosing the smallest exponent for each of them.

• In our example with 20 and 24, the common prime factor is 2. In 20, 2 appears as \( 2^2 \), and in 24, it appears as \( 2^3 \).
• The least power of 2 from these is \( 2^2 \) because it's smaller than \( 2^3 \).

Using the least power of common factors allows you to derive the greatest common factor efficiently, ensuring you use the lowest possible shared exponent.