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102 LearnersLast updated on September 18, 2025
GCF of 84 and 56
The GCF is the largest number that can divide two or more numbers without leaving any remainder. GCF is used to share the items equally, to group or arrange items, and schedule events. In this topic, we will learn about the GCF of 84 and 56.
What is the GCF of 84 and 56?
The greatest common factor of 84 and 56 is 28. 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 divisors are always positive.
How to find the GCF of 84 and 56?
To find the GCF of 84 and 56, a few methods are described below
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 84 and 56 by Using Listing of factors
Steps to find the GCF of 84 and 56 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 56 = 1, 2, 4, 7, 8, 14, 28, 56.
Step 2: Now, identify the common factors of them Common factors of 84 and 56: 1, 2, 4, 7, 14, 28.
Step 3: Choose the largest factor The largest factor that both numbers have is 28. The GCF of 84 and 56 is 28.
GCF of 84 and 56 Using Prime Factorization
To find the GCF of 84 and 56 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number
Prime Factors of 84: 84 = 2 x 2 x 3 x 7 = 2² x 3 x 7
Prime Factors of 56: 56 = 2 x 2 x 2 x 7 = 2³ x 7
Step 2: Now, identify the common prime factors The common prime factors are: 2 x 2 x 7 = 2² x 7
Step 3: Multiply the common prime factors 2² x 7 = 4 x 7 = 28. The Greatest Common Factor of 84 and 56 is 28.
GCF of 84 and 56 Using Division Method or Euclidean Algorithm Method
Find the GCF of 84 and 56 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number Here, divide 84 by 56 84 ÷ 56 = 1 (quotient),
The remainder is calculated as 84 − (56×1) = 28
The remainder is 28, not zero, so continue the process
Step 2: Now divide the previous divisor (56) by the previous remainder (28) Divide 56 by 28 56 ÷ 28 = 2 (quotient), remainder = 56 − (28×2) = 0
The remainder is zero, the divisor will become the GCF.
The GCF of 84 and 56 is 28.
Common Mistakes and How to Avoid Them in GCF of 84 and 56
Finding the GCF of 84 and 56 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 tells about the incomplete understanding of the factors. Students should include 1 as a factor.
Mistake 4
Using Multiples instead of factors
Students confuse between factors and 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 56 Examples
Problem 1
A baker has 84 chocolate cookies and 56 vanilla cookies. He wants to pack them into equal boxes with the largest number of cookies in each box. How many cookies will be in each box?
We should find the GCF of 84 and 56 GCF of 84 and 56 2² x 7 = 4 x 7 = 28.
There are 28 equal boxes 84 ÷ 28 = 3 56 ÷ 28 = 2
There will be 28 boxes, and each box gets 3 chocolate cookies and 2 vanilla cookies.
Explanation
As the GCF of 84 and 56 is 28, the baker can make 28 boxes.
Now divide 84 and 56 by 28.
Each box gets 3 chocolate cookies and 2 vanilla cookies.
Problem 2
A florist has 84 roses and 56 tulips. She wants to arrange them in vases with the same number of flowers in each vase, using the largest possible number of flowers per vase. How many flowers will be in each vase?
GCF of 84 and 56 2² x 7 = 4 × 7 = 28. So each vase will have 28 flowers.
Explanation
There are 84 roses and 56 tulips.
To find the total number of flowers in each vase, we should find the GCF of 84 and 56.
There will be 28 flowers in each vase.
Problem 3
A tailor has 84 meters of silk fabric and 56 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 56
The GCF of 84 and 56 2² x 7 = 4 × 7 = 28.
Each piece of fabric is 28 meters long.
Explanation
For calculating the longest length of the fabric, first, we need to calculate the GCF of 84 and 56 which is 28.
The length of each piece of fabric will be 28 meters.
Problem 4
A carpenter has two wooden planks, one 84 cm long and the other 56 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 56 2² x 7 = 4 × 7 = 28.
The longest length of each piece is 28 cm.
Explanation
To find the longest length of each piece of the two wooden planks, 84 cm and 56 cm, respectively, we have to find the GCF of 84 and 56, which is 28 cm.
The longest length of each piece is 28 cm.
Problem 5
If the GCF of 84 and ‘b’ is 28, and the LCM is 168. Find ‘b’.
The value of ‘b’ is 56.
Explanation
GCF x LCM = product of the numbers
28 × 168 = 84 × b
4704 = 84b
b = 4704 ÷ 84 = 56
FAQs on the Greatest Common Factor of 84 and 56
1.What is the LCM of 84 and 56?
2.Is 84 divisible by 3?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 56?
5.Are 84 and 56 prime numbers?
Important Glossaries for GCF of 84 and 56
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 28 are 1, 2, 4, 7, 14, and 28.
- Multiple: Multiples are the products we get by multiplying a given number by another. For example, the multiples of 7 are 7, 14, 21, 28, 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 42 are 2, 3, and 7.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 10 is divided by 3, the remainder is 1 and the quotient is 3.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 8 and 12 is 24.
- GCF: The largest factor that commonly divides two or more numbers. For example, the GCF of 14 and 28 will be 14, as it is their largest common factor that divides the numbers completely.
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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.
