The greatest common factor (GCF) is the greatest number that is a factor of two or more numbers. The first step for finding a GCF is to list the factors of the numbers. Then we can find which one is the greatest.
Given Numbers
Factors
GCF
39
1, 3, 13, 39
3
54
1, 2, 3, 6, 9, 18, 27, 54
3
63
1, 3, 7, 9, 21, 63
3
The GCF of 39, 54, and 63 is 3.
Alternative Solution
Finding the GCF Using Prime Factorization
We can also find the GCF using prime factorization. To do so, we will follow 3 steps.
Write the prime factorization for the given numbers.
Mark the common prime factors.
Multiply the marked factors. The product is the GCF.
Let's begin by finding the prime factorizations! The prime factorization of a composite number is the number written as a product of its prime factors. We will use factor pairs and a factor tree to find the prime factorization of 39, 54, and 63. The factor tree is complete when only prime factors appear in the product. Let's start with 39!
We found that the prime factorization of 39 is 3 * 13. Let's repeat the process with 54.
The prime factorization of the composite number 54 is 2 * 3 * 3 * 3. Let's continue with 63.
The prime factorization of 63 is 3 * 3* 7.
39 &= 3 * 13
54 &= 2 * 3 * 3 * 3
63 & = 3 * 3 * 7
The only common prime factor of 39, 54, and 63 is 3.
39 &= 3 * 13
54 &= 2 * 3 * 3 * 3
63 & = 3 * 3 * 7
The product of the marked factors is the GCF of the given numbers. In this case this is just 3.
GCF: 3
This method led us to the same answer, so we know the answer is reasonable.
