What are Common Factors?
The factor of a number divides the number completely without leaving any remainder. So, if we divide a number by its factor then the remainder will be zero, and the quotient obtained in this operation will also be a factor of the number.
When two or more numbers are divisible by the same factor then this factor is known as a common factor of the numbers. There can be one or more than one common factor, but the number 1 is the common factor of all numbers.
For example, common factors of 20 and 15 are 1 and 5.
How do we find Common Factors?
We can find the common factors by using two methods:
- Listing of Factors
In this method, we list the factors of each number and find the common factors from the list.
- Prime Factorization
In this method, we use a factor tree to find the prime factorization of each number and list out the common prime factors.
What is GCF?
GCF stands for greatest common factor, it is also known as the highest common factor(HCF). GCF is the greatest among the common factors of two or more numbers.
Solved Examples
Example 1: Find the common factor of 24, 28, and 34.
Solution:
List the factors of the given numbers:
So, the common factors of 24, 28, and 34 are 1 and 2.
Example 2: What is the greatest common factor of 18, 42, and 78?
Solution:
Let’s use the factor tree method to find the prime factors of 18, 42, and 78:
Write the prime factorization of each number.
18 = 2 x 3 x 3
42 = 2 x 3 x 7
78 = 2 x 3 x 13
So, the common factors are 2, 3, and 2 x 3 = 6.
The greatest of these common factors is 6.
Hence, the GCF of 18, 42 and 78 is 6.
Example 3: Olivia has 15 burgers and 25 pizzas. She wants to prepare identical food plates for a small party. She also doesn’t want to have any food left over. What is the maximum number of food plates that Olivia can prepare?
Solution:
The GCF of the number of each type of food item represents the maximum number of identical food plates Olivia can prepare with no food left over.
So, we will find the GCF of 15 and 25.
List the factors of the given numbers:
The common factors of 15 and 25 are 1 and 5.
The greatest of these common factors is 5.
Hence, the GCF of 15 and 25 is 5.
So, Olivia can prepare a maximum of 5 food plates with no food leftover.
The factors of a number are numbers that divide it completely without a remainder.
GCF or the greatest common factor is the greatest of all common factors of two or more numbers.
When a number is multiplied by another number, the product is known as a multiple of both the numbers and the numbers that are multiplied to each other are known as the factors of the product.
Even numbers are divisible by 1 as well as 2. So, 1 and 2 are always the common factors of even numbers.