Summarize this article:
101 LearnersLast updated on September 18, 2025
GCF of 21 and 30
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 schedule events. In this topic, we will learn about the GCF of 21 and 30.
What is the GCF of 21 and 30?
The greatest common factor of 21 and 30 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 21 and 30?
To find the GCF of 21 and 30, a few methods are described below
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 21 and 30 by Using Listing of Factors
Steps to find the GCF of 21 and 30 using the listing of factors
Step 1: Firstly, list the factors of each number
Factors of 21 = 1, 3, 7, 21.
Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30.
Step 2: Now, identify the common factors of them Common factors of 21 and 30: 1, 3.
Step 3: Choose the largest factor The largest factor that both numbers have is 3. The GCF of 21 and 30 is 3.
GCF of 21 and 30 Using Prime Factorization
To find the GCF of 21 and 30 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number
Prime Factors of 21: 21 = 3 x 7
Prime Factors of 30: 30 = 2 x 3 x 5
Step 2: Now, identify the common prime factors The common prime factor is: 3
Step 3: Multiply the common prime factors The Greatest Common Factor of 21 and 30 is 3.
GCF of 21 and 30 Using Division Method or Euclidean Algorithm Method
Find the GCF of 21 and 30 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number Here, divide 30 by 21 30 ÷ 21 = 1 (quotient),
The remainder is calculated as 30 − (21×1) = 9 The remainder is 9, not zero, so continue the process
Step 2: Now divide the previous divisor (21) by the previous remainder (9)
Divide 21 by 9 21 ÷ 9 = 2 (quotient), remainder = 21 − (9×2) = 3
The remainder is 3, not zero, so continue the process
Step 3: Now divide the previous divisor (9) by the previous remainder (3)
Divide 9 by 3 9 ÷ 3 = 3 (quotient), remainder = 9 − (3×3) = 0
The remainder is zero, the divisor will become the GCF. The GCF of 21 and 30 is 3.
Common Mistakes and How to Avoid Them in GCF of 21 and 30
Finding GCF of 21 and 30 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 21, students may mention 4 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 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; 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 21 and 30 Examples
Problem 1
A gardener has 21 rose bushes and 30 tulip plants. She wants to arrange them into equal rows with the largest number of plants in each row. How many plants will be in each row?
We should find the GCF of 21 and 30 GCF of 21 and 30 is 3.
There are 3 equal rows 21 ÷ 3 = 7 30 ÷ 3 = 10
There will be 3 rows, and each row gets 7 rose bushes and 10 tulip plants.
Explanation
As the GCF of 21 and 30 is 3, the gardener can make 3 rows.
Now divide 21 and 30 by 3.
Each row gets 7 rose bushes and 10 tulip plants.
Problem 2
A library has 21 fiction books and 30 non-fiction books. They want to organize them on shelves with the same number of books on each shelf, using the largest possible number of books per shelf. How many books will be on each shelf?
GCF of 21 and 30 is 3. So each shelf will have 3 books.
Explanation
There are 21 fiction and 30 non-fiction books.
To find the total number of books on each shelf, we should find the GCF of 21 and 30.
There will be 3 books on each shelf.
Problem 3
A tailor has 21 meters of green fabric and 30 meters of blue 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 21 and 30
The GCF of 21 and 30 is 3.
The fabric is 3 meters long.
Explanation
For calculating the longest length of the fabric, first, we need to calculate the GCF of 21 and 30, which is 3.
The length of each piece of the fabric will be 3 meters.
Problem 4
A carpenter has two wooden planks, one 21 cm long and the other 30 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 21 and 30 is 3.
The longest length of each piece is 3 cm.
Explanation
To find the longest length of each piece of the two wooden planks, 21 cm and 30 cm, respectively, we have to find the GCF of 21 and 30, which is 3 cm.
The longest length of each piece is 3 cm.
Problem 5
If the GCF of 21 and ‘b’ is 3, and the LCM is 210. Find ‘b’.
The value of ‘b’ is 30.
Explanation
GCF x LCM = product of the numbers
3 × 210 = 21 × b
630 = 21b
b = 630 ÷ 21 = 30
FAQs on the Greatest Common Factor of 21 and 30
1.What is the LCM of 21 and 30?
2.Is 21 divisible by 3?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 30?
5.Are 21 and 30 prime numbers?
Important Glossaries for GCF of 21 and 30
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 15 are 1, 3, 5, and 15.
- 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 18 are 2 and 3.
- 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 21 and 30 is 210.
Explore More numbers
![Important Math Links Icon]()
Previous to GCF of 21 and 30
![Important Math Links Icon]()
Next to GCF of 21 and 30
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.
