Determining whether two numbers have a greatest common factor (GCF) is an important concept in mathematics. The GCF of two or more numbers is the largest number that divides evenly into each of the numbers. Finding the GCF involves factoring the numbers completely and looking for common factors. Once the common factors are found, the largest of these factors is the GCF.
Factoring 52 and 39
To determine if 52 and 39 have a GCF, we first need to factor each number completely into prime factors. This involves breaking down each number into the product of prime numbers.
Let’s start with 52:
52 = 2 x 26
26 = 2 x 13
So the prime factorization of 52 is:
52 = 2 x 2 x 13
Now let’s factor 39:
39 = 3 x 13
We can see that 39 only has prime factors of 3 and 13. There are no other ways to factor it.
So the prime factorization of 39 is:
39 = 3 x 13
Finding Common Factors
Now that we have completely factored 52 and 39 into their prime factors, we can look for common factors shared between the two numbers.
The prime factors of 52 are: 2, 2, 13
The prime factors of 39 are: 3, 13
The only common prime factor between 52 and 39 is 13. Therefore, 13 is the greatest common factor (GCF) of 52 and 39.
Confirming with the GCF Formula
We can confirm that 13 is indeed the GCF of 52 and 39 by using the formula:
GCF = Product of common prime factors
Since the only common prime factor between 52 and 39 is 13, the GCF is:
GCF = 13
Therefore, by factoring 52 and 39 into their prime factors, identifying the common factor of 13, and confirming with the formula, we have shown that 13 is the greatest common factor of 52 and 39.
Visualizing with a Venn Diagram
A Venn diagram provides a visual representation of the common factors between two numbers. Here is a Venn diagram showing the prime factors of 52 and 39:

The overlap in the center representing 13 visually confirms that 13 is the GCF of 52 and 39.
GCF Properties and Examples
Understanding some key properties and looking at more examples helps solidify the concept of finding greatest common factors:
Properties of GCFs
 The GCF of two or more numbers is the largest factor that divides all the numbers exactly.
 The GCF of a set of numbers must be a factor of every number in the set.
 The GCF of a set of numbers cannot be larger than the smallest number in the set.
 The numbers 1 and the number itself are factors of every number. But the GCF is the largest common factor.
Examples
Find the GCF of: 12 and 30
Prime factors of 12: 2 x 2 x 3
Prime factors of 30: 2 x 3 x 5
Common factor = 6
Therefore, the GCF of 12 and 30 is 6.
Find the GCF of: 18, 24, and 36
Prime factors of 18: 2 x 3 x 3
Prime factors of 24: 2 x 2 x 2 x 3
Prime factors of 36: 2 x 2 x 3 x 3
Common factor = 6
Therefore, the GCF of 18, 24, and 36 is 6.
Finding the GCF of Polynomials
We can also find the GCF of polynomials by factoring them into their irreducible polynomial factors and looking for common factors.
For example:
Find the GCF of x^{3} + 4x + 4 and x^{2} + 10x + 16
Factoring:
x^{3} + 4x + 4 = (x + 2)^{2}(x + 2)
x^{2} + 10x + 16 = (x + 2)^{2}(x + 4)
The common factor is (x + 2).
Therefore, the GCF is x + 2.
Applications of GCF
Finding greatest common factors has many important applications in mathematics:
Simplifying Fractions
GCFs can be used to reduce fractions to their simplest forms. To simplify a fraction, divide the numerator and denominator by their GCF.
For example:
Simplify 36/60
The GCF of 36 and 60 is 12.
Divide the numerator and denominator by 12:
36/60 = 3/5
Solving Proportions
GCFs are useful when solving proportions. To keep a proportion equality, both sides must be multiplied or divided by the same value. Using the GCF maintains the equality.
For example:
2/3 = 4/x
Multiply both sides by 3 to clear the fractions:
2 = 4x/3
x = 2/4 = 1/2
Adding and Subtracting Fractions
When adding and subtracting fractions, using the GCF as a common denominator simplifies the process.
For example:
2/3 + 3/4
The GCF is 12. So convert the fractions:
8/12 + 9/12 = 17/12
Conclusion
In summary, determining if two numbers have a greatest common factor involves factoring them completely into prime factors, identifying any shared factors, and taking the largest of these shared factors as the GCF. We factors 52 and 39 into primes, found they shared the factor 13, and concluded 13 is their GCF. Venn diagrams and examples help visualize GCF concepts. Finding GCFs has many important applications like simplifying fractions, solving proportions, and working with fractions. Being able to determine greatest common factors provides an important skill for higher math and problem solving.