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101 LearnersLast updated on September 11, 2025
GCF of 6 and 21
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 6 and 21.
What is the GCF of 6 and 21?
The greatest common factor of 6 and 21 is 3. 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 6 and 21?
To find the GCF of 6 and 21, a few methods are described below:
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 6 and 21 by Using Listing of Factors
Steps to find the GCF of 6 and 21 using the listing of factors:
Step 1: Firstly, list the factors of each number.
Factors of 6 = 1, 2, 3, 6.
Factors of 21 = 1, 3, 7, 21.
Step 2: Now, identify the common factors of them. Common factors of 6 and 21: 1, 3.
Step 3: Choose the largest factor. The largest factor that both numbers have is 3. The GCF of 6 and 21 is 3.
GCF of 6 and 21 Using Prime Factorization
To find the GCF of 6 and 21 using Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number.
Prime factors of 6: 6 = 2 x 3
Prime factors of 21: 21 = 3 x 7
Step 2: Now, identify the common prime factors. The common prime factor is: 3
Step 3: Multiply the common prime factors. 3 = 3. The Greatest Common Factor of 6 and 21 is 3.
GCF of 6 and 21 Using Division Method or Euclidean Algorithm Method
Find the GCF of 6 and 21 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number. Here, divide 21 by 6. 21 ÷ 6 = 3 (quotient), The remainder is calculated as 21 - (6×3) = 3. The remainder is 3, not zero, so continue the process.
Step 2: Now divide the previous divisor (6) by the previous remainder (3). Divide 6 by 3. 6 ÷ 3 = 2 (quotient), remainder = 6 - (3×2) = 0.
The remainder is zero, the divisor will become the GCF. The GCF of 6 and 21 is 3.
Common Mistakes and How to Avoid Them in GCF of 6 and 21
Finding GCF of 6 and 21 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 6, 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 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 6 and 21 Examples
Problem 1
A farmer has 6 apple trees and 21 orange trees. He wants to plant them in rows with the same number of trees in each row. How many trees will be in each row?
We should find GCF of 6 and 21. GCF of 6 and 21 3.
There are 3 equal groups. 6 ÷ 3 = 2
21 ÷ 3 = 7
There will be 3 rows, and each row gets 2 apple trees and 7 orange trees.
Explanation
As the GCF of 6 and 21 is 3, the farmer can make 3 rows. Now divide 6 and 21 by 3. Each row gets 2 apple trees and 7 orange trees.
Problem 2
A chef has 6 red bowls and 21 blue bowls. He wants to arrange them in stacks with the same number of bowls in each stack, using the largest possible number of bowls per stack. How many bowls will be in each stack?
GCF of 6 and 21 3. So each stack will have 3 bowls.
Explanation
There are 6 red bowls and 21 blue bowls. To find the total number of bowls in each stack, we should find the GCF of 6 and 21. There will be 3 bowls in each stack.
Problem 3
A landscaper has 6 meters of red fencing and 21 meters of green fencing. She wants to cut both fences 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 6 and 21. The GCF of 6 and 21 3. The fencing is 3 meters long.
Explanation
For calculating the longest length of the fencing first we need to calculate the GCF of 6 and 21 which is 3. The length of each piece of fencing will be 3 meters.
Problem 4
A carpenter has two wooden planks, one 6 cm long and the other 21 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 6 and 21 3. The longest length of each piece is 3 cm.
Explanation
To find the longest length of each piece of the two wooden planks, 6 cm and 21 cm, respectively. We have to find the GCF of 6 and 21, which is 3 cm. The longest length of each piece is 3 cm.
Problem 5
If the GCF of 6 and ‘b’ is 3, and the LCM is 42. Find ‘b’.
The value of ‘b’ is 21.
Explanation
GCF x LCM = product of the numbers 3 × 42 = 6 × b 126 = 6b b = 126 ÷ 6 = 21
FAQs on the Greatest Common Factor of 6 and 21
1.What is the LCM of 6 and 21?
2.Is 6 divisible by 2?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 21?
5.Are 6 and 21 prime numbers?
Important Glossaries for GCF of 6 and 21
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 6 are 1, 2, 3, and 6.
- Multiple: Multiples are the products we get by multiplying a given number by another. For example, the multiples of 3 are 3, 6, 9, 12, 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 21 are 3 and 7.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 21 is divided by 6, the remainder is 3 and the quotient is 3.
- GCF: The largest factor that commonly divides two or more numbers. For example, the GCF of 6 and 21 is 3, 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.
