What is the greatest common factor of 28 and 72?
Understand the Problem
The question is asking to find the greatest common factor (GCF) of the numbers 28 and 72. The GCF is the largest number that can divide both integers without leaving a remainder. To solve this, we will find the prime factorization of both numbers and then determine the common factors.
Answer
4
Answer for screen readers
The greatest common factor (GCF) of 28 and 72 is $4$.
Steps to Solve
- Find the Prime Factorization of 28
To find the GCF, we first need to break down 28 into its prime factors.
28 can be factored as: $$ 28 = 2 \times 14 $$ Next, factor 14: $$ 14 = 2 \times 7 $$ So, the prime factorization of 28 is: $$ 28 = 2^2 \times 7 $$
- Find the Prime Factorization of 72
Now, let's break down 72 into its prime factors.
72 can be factored as: $$ 72 = 8 \times 9 $$ Next, factor 8 and 9: $$ 8 = 2^3 \quad \text{and} \quad 9 = 3^2 $$ So, the prime factorization of 72 is: $$ 72 = 2^3 \times 3^2 $$
- Identify Common Prime Factors
Now we will look for common prime factors between 28 and 72. The prime factorization results are:
- For 28: $2^2 \times 7$
- For 72: $2^3 \times 3^2$
The only common prime factor is 2.
- Find the Lowest Power of Common Factors
Next, we take the lowest power of the common prime factor.
The lowest power of 2 in the prime factorizations is $2^2$.
- Calculate the GCF
So, the GCF is: $$ GCF = 2^2 = 4 $$
The greatest common factor (GCF) of 28 and 72 is $4$.
More Information
The GCF is useful in various mathematical applications such as simplifying fractions, finding common denominators, and solving problems involving ratios. Understanding GCF can help in reducing calculations to their simplest form.
Tips
- Ignoring Non-Common Factors: When finding the GCF, ensure to only consider the common prime factors. Do not include any factors that appear in only one of the numbers.
- Incorrectly Summarizing Prime Factorization: Make sure to accurately express the complete prime factorization of each number. Missing any factors can lead to an inaccurate GCF.
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