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103 LearnersLast updated on September 24, 2025
GCF of 84 and 128
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 84 and 128.
What is the GCF of 84 and 128?
The greatest common factor of 84 and 128 is 4. The largest divisor of two or more numbers is called the GCF of the number.
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 a divisor is always positive.
How to find the GCF of 84 and 128?
To find the GCF of 84 and 128, a few methods are described below
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 84 and 128 by Using Listing of Factors
Steps to find the GCF of 84 and 128 using the listing of factors:
Step 1: Firstly, list the factors of each number Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. Factors of 128 = 1, 2, 4, 8, 16, 32, 64, 128.
Step 2: Now, identify the common factors Common factors of 84 and 128: 1, 2, 4.
Step 3: Choose the largest factor The largest factor that both numbers have is 4. The GCF of 84 and 128 is 4.
GCF of 84 and 128 Using Prime Factorization
To find the GCF of 84 and 128 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number Prime Factors of 84: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7 Prime Factors of 128: 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁷
Step 2: Now, identify the common prime factors The common prime factors are: 2 × 2 = 2²
Step 3: Multiply the common prime factors 2² = 4. The Greatest Common Factor of 84 and 128 is 4.
GCF of 84 and 128 Using Division Method or Euclidean Algorithm Method
Find the GCF of 84 and 128 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number Here, divide 128 by 84 128 ÷ 84 = 1 (quotient), The remainder is calculated as 128 − (84×1) = 44 The remainder is 44, not zero, so continue the process
Step 2: Now divide the previous divisor (84) by the previous remainder (44) Divide 84 by 44 84 ÷ 44 = 1 (quotient), remainder = 84 − (44×1) = 40
Step 3: Continue dividing 44 ÷ 40 = 1 (quotient), remainder = 44 − (40×1) = 4
Step 4: Divide 40 by 4 40 ÷ 4 = 10 (quotient), remainder = 40 − (4×10) = 0 The remainder is zero, so the divisor will become the GCF.
The GCF of 84 and 128 is 4.
Common Mistakes and How to Avoid Them in GCF of 84 and 128
Finding the GCF of 84 and 128 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 84, students may mention 5, 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 an 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 that 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 84 and 128 Examples
Problem 1
A gardener has 84 tulips and 128 roses. She wants to plant them in rows with the largest possible number of flowers in each row. How many flowers will be in each row?
We should find the GCF of 84 and 128. GCF of 84 and 128 2² = 4. There are 4 flowers in each row. 84 ÷ 4 = 21 128 ÷ 4 = 32 There will be 4 rows, with each row containing 21 tulips and 32 roses.
Explanation
As the GCF of 84 and 128 is 4, the gardener can make rows with 4 flowers each.
Now divide 84 and 128 by 4.
Each row will have 21 tulips and 32 roses.
Problem 2
A school has 84 desks and 128 chairs. They want to arrange them in groups with the same number of desks and chairs in each group, using the greatest possible number of items per group. How many items will be in each group?
GCF of 84 and 128 2² = 4. So each group will have 4 items.
Explanation
There are 84 desks and 128 chairs.
To find the total number of items in each group, we should find the GCF of 84 and 128.
There will be 4 items in each group.
Problem 3
A tailor has 84 meters of silk fabric and 128 meters of cotton 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 84 and 128. The GCF of 84 and 128 2² = 4. The fabric pieces are 4 meters long.
Explanation
For calculating the longest length of each fabric piece, first we need to calculate the GCF of 84 and 128, which is 4.
The length of each piece of fabric will be 4 meters.
Problem 4
A carpenter has two wooden planks, one 84 cm long and the other 128 cm long. He wants to cut them into the longest possible equal pieces, without any wood left over. What should be the length of each piece?
The carpenter needs the longest piece of wood. GCF of 84 and 128 2² = 4. The longest length of each piece is 4 cm.
Explanation
To find the longest length of each piece of the two wooden planks, 84 cm and 128 cm, respectively, we have to find the GCF of 84 and 128, which is 4 cm.
The longest length of each piece is 4 cm.
Problem 5
If the GCF of 84 and ‘b’ is 4, and the LCM is 2688, find ‘b’.
The value of ‘b’ is 128.
Explanation
GCF × LCM = product of the numbers
4 × 2688 = 84 × b
10752 = 84b
b = 10752 ÷ 84
= 128
FAQs on the Greatest Common Factor of 84 and 128
1.What is the LCM of 84 and 128?
2.Is 128 divisible by 2?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 84?
5.Are 84 and 128 prime numbers?
Important Glossaries for GCF of 84 and 128
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 4 are 1, 2, and 4.
- 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 84 are 2, 3, and 7.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 44 is divided by 40, the remainder is 4 and the quotient is 1.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 84 and 128 is 2688.
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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.
