Summarize this article:
104 LearnersLast updated on October 3, 2025
GCF of 10 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 10 and 21.
What is the GCF of 10 and 21?
The greatest common factor of 10 and 21 is 1.
The largest divisor of two or more numbers is called the GCF of the numbers.
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 10 and 21?
To find the GCF of 10 and 21, a few methods are described below -
Listing Factors Prime Factorization Long Division Method / Euclidean Algorithm
GCF of 10 and 21 by Using Listing of factors
Steps to find the GCF of 10 and 21 using the listing of factors
Step 1: Firstly, list the factors of each number Factors of 10 = 1, 2, 5, 10. Factors of 21 = 1, 3, 7, 21.
Step 2: Now, identify the common factors of them Common factor of 10 and 21: 1.
Step 3: Choose the largest factor, The largest factor that both numbers have is 1.
The GCF of 10 and 21 is 1.
GCF of 10 and 21 Using Prime Factorization
To find the GCF of 10 and 21 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number Prime Factors of 10: 10 = 2 x 5 Prime Factors of 21: 21 = 3 x 7
Step 2: Now, identify the common prime factors There are no common prime factors.
Step 3: Since there are no common prime factors, the GCF is 1.
The Greatest Common Factor of 10 and 21 is 1.
GCF of 10 and 21 Using Division Method or Euclidean Algorithm
Find the GCF of 10 and 21 using the division method or Euclidean Algorithm.
Follow these steps:
Step 1: First, divide the larger number by the smaller number Here, divide 21 by 10 21 ÷ 10 = 2 (quotient), The remainder is calculated as 21 − (10×2) = 1 The remainder is 1, not zero, so continue the process.
Step 2: Now divide the previous divisor (10) by the previous remainder (1) Divide 10 by 1 10 ÷ 1 = 10 (quotient), remainder = 10 − (1×10) = 0, The remainder is zero, the divisor will become the GCF.
The GCF of 10 and 21 is 1.
Common Mistakes and How to Avoid Them in GCF of 10 and 21
Finding the GCF of 10 and 21 seems 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 10, students may mention 8 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 shows 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; 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 10 and 21 Examples
Problem 1
A teacher has 10 apples and 21 oranges. She wants to group them into equal sets, with the largest number of items in each group. How many items will be in each group?
We should find the GCF of 10 and 21. GCF of 10 and 21 is 1.
There are 1 equal groups. 10 ÷ 1 = 10 21 ÷ 1 = 21.
There will be 1 group, and each group gets 10 apples and 21 oranges.
Explanation
As the GCF of 10 and 21 is 1, the teacher can make 1 group. Now divide 10 and 21 by 1.
Each group gets 10 apples and 21 oranges.
Problem 2
A school has 10 red chairs and 21 blue chairs. They want to arrange them in rows with the same number of chairs in each row, using the largest possible number of chairs per row. How many chairs will be in each row?
GCF of 10 and 21 is 1.
So each row will have 1 chair.
Explanation
There are 10 red and 21 blue chairs.
To find the total number of chairs in each row, we should find the GCF of 10 and 21.
There will be 1 chair in each row.
Problem 3
A tailor has 10 meters of red ribbon and 21 meters of blue ribbon. She wants to cut both ribbons 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 10 and 21.
The GCF of 10 and 21 is 1.
The ribbon is 1 meter long.
Explanation
For calculating the longest length of the ribbon, first we need to calculate the GCF of 10 and 21, which is 1.
The length of each piece of the ribbon will be 1 meter.
Problem 4
A carpenter has two wooden planks, one 10 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 10 and 21 is 1.
The longest length of each piece is 1 cm.
Explanation
To find the longest length of each piece of the two wooden planks, 10 cm and 21 cm, respectively, we have to find the GCF of 10 and 21, which is 1 cm.
The longest length of each piece is 1 cm.
Problem 5
If the GCF of 10 and ‘a’ is 1, and the LCM is 210. Find ‘a’.
The value of ‘a’ is 21.
Explanation
GCF x LCM = product of the numbers 1 × 210 = 10 × a 210 = 10a a = 210 ÷ 10 = 21
FAQs on the Greatest Common Factor of 10 and 21
1.What is the LCM of 10 and 21?
2.Is 10 divisible by 2?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 21?
5.Are 10 and 21 prime numbers?
Important Glossaries for GCF of 10 and 21
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 10 are 1, 2, 5, and 10.
- 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 10, the remainder is 1 and the quotient is 2.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 10 and 21 is 210.
Explore More numbers
![Important Math Links Icon]()
Previous to GCF of 10 and 21
![Important Math Links Icon]()
Next to GCF of 10 and 21
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.
