Summarize this article:
102 LearnersLast updated on September 9, 2025
GCF of 96 and 84
The GCF is the largest number that can divide two or more numbers without leaving any remainder. GCF is used to share items equally, to group or arrange items, and to schedule events. In this topic, we will learn about the GCF of 96 and 84.
What is the GCF of 96 and 84?
The greatest common factor of 96 and 84 is 12. The largest divisor of two or more numbers is called the GCF of the numbers.
If two numbers are co-prime, they have no common factors other than 1, so their GCF is 1. The GCF of two numbers cannot be negative because divisors are always positive.
How to find the GCF of 96 and 84?
To find the GCF of 96 and 84, a few methods are described below
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 96 and 84 by Using Listing of factors
Steps to find the GCF of 96 and 84 using the listing of factors
Step 1: Firstly, list the factors of each number Factors of 96 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96. Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
Step 2: Now, identify the common factors of them Common factors of 96 and 84: 1, 2, 3, 4, 6, 12.
Step 3: Choose the largest factor The largest factor that both numbers have is 12. The GCF of 96 and 84 is 12.
GCF of 96 and 84 Using Prime Factorization
To find the GCF of 96 and 84 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number Prime Factors of 96: 96 = 2 × 2 × 2 × 2 × 2 × 3 = 2^5 × 3 Prime Factors of 84: 84 = 2 × 2 × 3 × 7 = 2^2 × 3 × 7
Step 2: Now, identify the common prime factors The common prime factors are: 2 × 2 × 3 = 2^2 × 3
Step 3: Multiply the common prime factors 2^2 × 3 = 4 × 3 = 12. The Greatest Common Factor of 96 and 84 is 12.
GCF of 96 and 84 Using Division Method or Euclidean Algorithm Method
Find the GCF of 96 and 84 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number Here, divide 96 by 84 96 ÷ 84 = 1 (quotient), remainder is calculated as 96 − (84×1) = 12 The remainder is 12, not zero, so continue the process
Step 2: Now divide the previous divisor (84) by the previous remainder (12) Divide 84 by 12 84 ÷ 12 = 7 (quotient), remainder = 84 − (12×7) = 0 The remainder is zero, the divisor will become the GCF. The GCF of 96 and 84 is 12.
Common Mistakes and How to Avoid Them in GCF of 96 and 84
Finding the GCF of 96 and 84 looks simple, but students often make mistakes while calculating the GCF. Here are some common mistakes to be avoided by the students.
Mistake 1
Listing Incorrect Factors
Students may sometimes list incorrect factors. For example, while listing factors of 96, students may mention 10, which is incorrect.
To avoid this, students should carefully divide the number and list the factors correctly.
Mistake 2
Choosing the wrong common factor
Students may sometimes select the smallest common factor instead of the largest one.
To avoid this confusion, students should list all the common factors and find the greatest one.
Mistake 3
Forgetting to include 1 as a factor
Sometimes students may forget 1 as a common factor of the numbers.
However, it does not affect the GCF, but it indicates incomplete understanding of the factors. Students should include 1 as a factor.
Mistake 4
Using Multiples instead of factors
Students confuse factors with multiples. In that confusion, sometimes they may write multiples instead of factors.
To avoid this confusion, students should know the definitions of multiples and factors clearly.
Mistake 5
Assuming GCF is always an even number
Students may assume that the GCF of two numbers will always be an even number. But it's not true; a GCF can also be an odd number.
To avoid this, students should focus on common factors rather than focusing on even and odd numbers.
Greatest Common Factor of 96 and 84 Examples
Problem 1
A chef has 96 apples and 84 oranges. She wants to pack them into equal sets, with the largest number of fruits in each set. How many fruits will be in each set?
We should find the GCF of 96 and 84. GCF of 96 and 84 2^2 × 3 = 4 × 3 = 12. There are 12 equal groups. 96 ÷ 12 = 8 84 ÷ 12 = 7 There will be 12 groups, and each group gets 8 apples and 7 oranges.
Explanation
As the GCF of 96 and 84 is 12, the chef can make 12 groups.
Now divide 96 and 84 by 12.
Each group gets 8 apples and 7 oranges.
Problem 2
A warehouse has 96 large boxes and 84 small boxes. They want to arrange them in rows with the same number of boxes in each row, using the largest possible number of boxes per row. How many boxes will be in each row?
GCF of 96 and 84 2^2 × 3 = 4 × 3 = 12. So each row will have 12 boxes.
Explanation
There are 96 large and 84 small boxes.
To find the total number of boxes in each row, we should find the GCF of 96 and 84.
There will be 12 boxes in each row.
Problem 3
A tailor has 96 meters of cotton fabric and 84 meters of silk fabric. She wants to cut both fabrics into pieces of equal length, using the longest possible length. What should be the length of each piece?
For calculating the longest equal length, we have to calculate the GCF of 96 and 84. The GCF of 96 and 84 2^2 × 3 = 4 × 3 = 12. The fabric is 12 meters long.
Explanation
For calculating the longest length of the fabric, first we need to calculate the GCF of 96 and 84, which is 12.
The length of each piece of the fabric will be 12 meters.
Problem 4
A gardener has two plots of land, one 96 square meters and the other 84 square meters. He wants to divide them into the largest possible equal sections, without any land left over. What should be the area of each section?
The gardener needs the largest section of land. GCF of 96 and 84 2^2 × 3 = 4 × 3 = 12. The largest area of each section is 12 square meters.
Explanation
To find the largest area of each section for the two plots, 96 square meters and 84 square meters, respectively, we have to find the GCF of 96 and 84, which is 12 square meters.
The largest area of each section is 12 square meters.
Problem 5
If the GCF of 96 and ‘b’ is 12, and the LCM is 672, find ‘b’.
The value of ‘b’ is 84.
Explanation
GCF × LCM = product of the numbers
12 × 672
= 96 × b 8064
= 96b b
= 8064 ÷ 96 = 84
FAQs on the Greatest Common Factor of 96 and 84
1.What is the LCM of 96 and 84?
2.Is 96 divisible by 2?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 84?
5.Are 96 and 84 prime numbers?
Important Glossaries for GCF of 96 and 84
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
- Multiple: Multiples are the products we get by multiplying a given number by another. For example, the multiples of 4 are 4, 8, 12, 16, 20, and so on.
- Prime Factors: These are the factors of a number that are prime numbers and divide the given number completely. For example, the prime factors of 15 are 3 and 5.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 12 is divided by 7, the remainder is 5 and the quotient is 1.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 96 and 84 is 672.
Explore More numbers
![Important Math Links Icon]()
Previous to GCF of 96 and 84
![Important Math Links Icon]()
Next to GCF of 96 and 84
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
