How to Find the GCF of Two Numbers Using Prime Factorization and Division Method
What Are The Factors?
In Mathematics, factors are those numbers which on multiplication gives us the original number. Factors are always whole numbers. Only the decimal numbers and fractions are not considered as factors. To find factors, you must be familiar with the multiplication table so it will be easy to find them. The numbers can be factorized into different combinations. There are various methods to find the factors of the number. They are the factorization method, the prime factorization method, division method, and the divisibility method. Here, we are going to discuss what is the greatest common factor and how to find it.
Greatest Common Factor
The factor which is common and largest between the two numbers is called the greatest common factor. It is the largest number which divides them giving the answer as the whole number.
The greatest common factor (GCF) is also known as the highest common factor (HCF).
For example, find the greatest common factor (GCF) of 12 and 20.
The Factors of 12 are 2 × 2 × 3
The Factors of 20 are 2 × 2 × 5
Then common factors are 2 so that the greatest common factor (GCF) of 12 and 20 is 2.
Example - Find the greatest common factor (GCF) of 20 and 30.
The Factors of 20 = 2 × 2 × 5
The Factors of 30 = 2 × 3 × 5
Here the common factors are 2 and 5 so the greatest common factor (GCF) of 20 and 30 is 2 × 5= 10.
Factoring Greatest Common Factor
In this, we will use the factorization method to list out all the factors of the given numbers. In the factorization method, the factors are found by considering two numbers which on multiplication gives us the original number.
By calculating factors, it will be easy to find the greatest common factor (GCF) and the least common multiple (LCM).
Example 1- Find the greatest common factor (GCF)of 12 and 18 using the factoring method.
Then Factors of 12 are 1, 2, 3, 4, 6, 12.
Because 1 × 12 = 12, 2 × 6 = 12, and 3 × 4 = 12
Then Factors of 18 are 1, 2, 3, 6, 9, 18.
Because 1 × 18 = 18, 2 × 9 = 18, and 3 × 6 = 19.
Thus, the common factors between 12 and 18 are 1, 2, 3, 6.
The largest common factor is 6.
Therefore, the greatest common factor (GCF) of 12 and 18 is 6.
Example-2
Find out the least common multiple (LCM) of 24 and 36.
Then Factors of 24 are 2 × 2 × 2 × 3.
Then Factors of 36 are 2 × 2 × 3 × 3.
A least common multiple (LCM) of 24 and 36 is 2 × 2 × 2 × 2 × 3 = 72.
Find the greatest common factor (GCF) of 8, 18,28, and 48.
Solution:
First, we will find factors of all the numbers,
Then Factors of 8 = 1, 2, 4, 8.
Then Factors of 18 = 1, 2, 3, 6, 9, 18.
Then Factors of 28 = 1, 2, 4, 7, 14, 28.
Then Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
Now we will find the common factors between these numbers.
The largest common factor of 8, 18, 28, 48 is 2. Because factors 1 and 2 are found to be the common factors of these numbers. Between 1 and 2 factors 2 is the largest common factor, therefore, the greatest common factor (GCF) is 2.
FAQs on GCF of Two Numbers Complete Guide with Methods
1. What is the GCF of two numbers?
The Greatest Common Factor (GCF) of two numbers is the largest number that divides both numbers exactly without leaving a remainder. It is also called the Greatest Common Divisor (GCD) or Highest Common Factor (HCF).
- It must be a common factor of both numbers.
- It must be the greatest among all common factors.
- Example: The GCF of 12 and 18 is 6.
2. How do you find the GCF of two numbers?
You can find the GCF of two numbers using listing factors, prime factorization, or the division (Euclidean) method.
- Step 1: List all factors of each number.
- Step 2: Identify common factors.
- Step 3: Choose the greatest common factor.
3. What is the formula for GCF using prime factorization?
The GCF using prime factorization is found by multiplying the common prime factors with the smallest powers.
- Step 1: Write each number as a product of prime factors.
- Step 2: Identify common prime factors.
- Step 3: Multiply the common primes with the least exponent.
4. What is the GCF of 12 and 18?
The GCF of 12 and 18 is 6.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common factors: 1, 2, 3, 6
5. What is the difference between GCF and LCM?
The GCF is the largest number that divides two numbers, while the LCM (Least Common Multiple) is the smallest number that both numbers divide into exactly.
- GCF deals with common factors.
- LCM deals with common multiples.
- Example for 4 and 6: GCF = 2, LCM = 12.
6. How do you find the GCF using the Euclidean algorithm?
The Euclidean algorithm finds the GCF by repeatedly dividing and taking remainders until the remainder is zero.
- Step 1: Divide the larger number by the smaller number.
- Step 2: Replace the larger number with the smaller number and the smaller number with the remainder.
- Step 3: Repeat until remainder = 0.
7. Can the GCF of two numbers be 1?
Yes, the GCF of two numbers can be 1 if they are coprime (relatively prime). This means they have no common factors other than 1.
- Example: 8 and 15
- Common factor: 1 only
- So, GCF = 1
8. What is the GCF of 24 and 36?
The GCF of 24 and 36 is 12.
- 24 = 2³ × 3
- 36 = 2² × 3²
- Common primes with smallest powers: 2² and 3¹
9. Why is the GCF important in math?
The GCF is important because it helps simplify fractions, factor algebraic expressions, and solve word problems.
- Used to reduce fractions to lowest terms.
- Used in factoring polynomials.
- Helps divide quantities into equal groups.
10. How do you use GCF to simplify fractions?
To simplify a fraction using the GCF, divide both the numerator and denominator by their greatest common factor.
- Step 1: Find the GCF of numerator and denominator.
- Step 2: Divide both by the GCF.
