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101 LearnersLast updated on September 24, 2025
GCF of 3 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, group or arrange items, and schedule events. In this topic, we will learn about the GCF of 3 and 21.
What is the GCF of 3 and 21?
The greatest common factor of 3 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 3 and 21?
To find the GCF of 3 and 21, a few methods are described below
- Listing Factors
- Prime Factorization
- Long Division Method / Euclidean Algorithm
GCF of 3 and 21 by Using Listing of Factors
Steps to find the GCF of 3 and 21 using the listing of factors
Step 1: Firstly, list the factors of each number Factors of 3 = 1, 3. Factors of 21 = 1, 3, 7, 21.
Step 2: Now, identify the common factors of them Common factors of 3 and 21: 1, 3.
Step 3: Choose the largest factor The largest factor that both numbers have is 3. The GCF of 3 and 21 is 3.
GCF of 3 and 21 Using Prime Factorization
To find the GCF of 3 and 21 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number Prime Factors of 3 : 3 = 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 3 and 21 is 3.
GCF of 3 and 21 Using Division Method or Euclidean Algorithm Method
Find the GCF of 3 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 3 21 ÷ 3 = 7 (quotient), The remainder is calculated as 21 − (3×7) = 0 The remainder is zero, so the divisor becomes the GCF.
The GCF of 3 and 21 is 3.
Common Mistakes and How to Avoid Them in GCF of 3 and 21
Finding the GCF of 3 and 21 looks simple, but students often make mistakes while calculating the GCF. Here are some common mistakes to be avoided by students.
Mistake 1
Listing Incorrect Factors
Students may sometimes list incorrect factors. For example, while listing factors of 21, 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 an incomplete understanding of the factors. Students should include 1 as a factor.
Mistake 4
Using Multiples Instead of Factors
Students confuse 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 Odd Number
Students may assume that the GCF of two numbers will always be an odd number. But it's not true; a GCF can also be an even number.
To avoid this, students should focus on common factors rather than focusing on even and odd numbers.
Greatest Common Factor of 3 and 21 Examples
Problem 1
A gardener has 3 rose plants and 21 tulip plants. She wants to plant them in rows with the same number of plants in each row. How many plants will be in each row?
We should find the GCF of 3 and 21 GCF of 3 and 21 3 = 3. There are 3 equal plants per row. 3 ÷ 3 = 1 21 ÷ 3 = 7 There will be 3 plants in each row, with 1 rose and 7 tulips.
Explanation
As the GCF of 3 and 21 is 3, the gardener can make rows with 3 plants.
Now divide 3 and 21 by 3.
Each row gets 1 rose plant and 7 tulip plants.
Problem 2
A farmer has 3 cows and 21 chickens. He wants to arrange them in groups with the same number of animals in each group. What is the maximum number of animals in each group?
GCF of 3 and 21 3 = 3. So each group will have 3 animals.
Explanation
There are 3 cows and 21 chickens.
To find the total number of animals in each group, we should find the GCF of 3 and 21.
Each group will have 3 animals.
Problem 3
A chef has 3 kg of sugar and 21 kg of flour. She wants to divide them into packages of equal weight, using the largest possible weight for each package. What should be the weight of each package?
For calculating the largest equal weight, we have to calculate the GCF of 3 and 21 The GCF of 3 and 21 3 = 3. Each package will weigh 3 kg.
Explanation
For calculating the largest weight of each package, first we need to calculate the GCF of 3 and 21, which is 3.
The weight of each package will be 3 kg.
Problem 4
A school has two sets of books, one with 3 science books and another with 21 math books. They want to distribute them into boxes with the same number of books in each box. What should be the number of books in each box?
The school needs the largest number of books per box GCF of 3 and 21 3 = 3. The number of books in each box is 3.
Explanation
To find the number of books in each box, we have to find the GCF of 3 and 21, which is 3.
The number of books in each box will be 3.
Problem 5
If the GCF of 3 and 'b' is 3, and the LCM is 21, find 'b'.
The value of 'b' is 21.
Explanation
GCF x LCM = product of the numbers
3 × 21 = 3 × b
63 = 3b
b = 63 ÷ 3
= 21
FAQs on the Greatest Common Factor of 3 and 21
1.What is the LCM of 3 and 21?
2.Is 3 a prime number?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 21?
5.Are 3 and 21 prime numbers?
Important Glossaries for GCF of 3 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, 15, 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 22 is divided by 5, the remainder is 2, and the quotient is 4.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 3 and 21 is 21.
- GCF: The largest factor that commonly divides two or more numbers. For example, the GCF of 9 and 15 will be 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.
