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104 LearnersLast updated on September 20, 2025
GCF of 120 and 210
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 to schedule events. In this topic, we will learn about the GCF of 120 and 210.
What is the GCF of 120 and 210?
The greatest common factor of 120 and 210 is 30. 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 120 and 210?
To find the GCF of 120 and 210, a few methods are described below:
- Listing Factors
- Prime Factorization
- Long Division Method / Euclidean Algorithm
GCF of 120 and 210 by Using Listing of factors
Steps to find the GCF of 120 and 210 using the listing of factors:
Step 1: Firstly, list the factors of each number
Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.
Factors of 210 = 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210.
Step 2: Now, identify the common factors of them Common factors of 120 and 210: 1, 2, 3, 5, 6, 10, 15, 30.
Step 3: Choose the largest factor The largest factor that both numbers have is 30.
The GCF of 120 and 210 is 30.
GCF of 120 and 210 Using Prime Factorization
To find the GCF of 120 and 210 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number
Prime Factors of 120: 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5
Prime Factors of 210: 210 = 2 × 3 × 5 × 7 = 2 × 3 × 5 × 7
Step 2: Now, identify the common prime factors The common prime factors are: 2 × 3 × 5 = 2 × 3 × 5
Step 3: Multiply the common prime factors 2 × 3 × 5 = 30.
The Greatest Common Factor of 120 and 210 is 30.
GCF of 120 and 210 Using Division Method or Euclidean Algorithm Method
Find the GCF of 120 and 210 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number
Here, divide 210 by 120 210 ÷ 120 = 1 (quotient),
The remainder is calculated as 210 − (120×1) = 90
The remainder is 90, not zero, so continue the process
Step 2: Now divide the previous divisor (120) by the previous remainder (90)
Divide 120 by 90 120 ÷ 90 = 1 (quotient), remainder = 120 − (90×1) = 30
Step 3: Continue with the new divisor (90) and the new remainder (30)
Divide 90 by 30 90 ÷ 30 = 3 (quotient), remainder = 90 − (30×3) = 0
The remainder is zero, so the divisor will become the GCF.
The GCF of 120 and 210 is 30.
Common Mistakes and How to Avoid Them in GCF of 120 and 210
Finding the GCF of 120 and 210 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 120, students may mention 25, 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 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 120 and 210 Examples
Problem 1
A farmer has 120 apples and 210 oranges. He wants to package them into equal sets, with the largest number of fruits in each package. How many fruits will be in each package?
We should find the GCF of 120 and 210 GCF of 120 and 210 2 × 3 × 5 = 30.
There are 30 equal sets 120 ÷ 30 = 4 210 ÷ 30 = 7
There will be 30 sets, and each set gets 4 apples and 7 oranges.
Explanation
As the GCF of 120 and 210 is 30, the farmer can make 30 sets.
Now divide 120 and 210 by 30.
Each set gets 4 apples and 7 oranges.
Problem 2
A school has 120 red chairs and 210 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 120 and 210 2 × 3 × 5 = 30.
So each row will have 30 chairs.
Explanation
There are 120 red and 210 blue chairs. To find the total number of chairs in each row, we should find the GCF of 120 and 210.
There will be 30 chairs in each row.
Problem 3
A tailor has 120 meters of red fabric and 210 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 120 and 210
The GCF of 120 and 210 2 × 3 × 5 = 30.
The fabric is 30 meters long.
Explanation
For calculating the longest length of the fabric, first we need to calculate the GCF of 120 and 210, which is 30.
The length of each piece of the fabric will be 30 meters.
Problem 4
A carpenter has two wooden planks, one 120 cm long and the other 210 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 120 and 210
2 × 3 × 5 = 30.
The longest length of each piece is 30 cm.
Explanation
To find the longest length of each piece of the two wooden planks, 120 cm and 210 cm, respectively, we have to find the GCF of 120 and 210, which is 30 cm.
The longest length of each piece is 30 cm.
Problem 5
If the GCF of 120 and ‘a’ is 30, and the LCM is 840, find ‘a’.
The value of ‘a’ is 210.
Explanation
GCF × LCM = product of the numbers
30 × 840 = 120 × a
25200 = 120a
a = 25200 ÷ 120 = 210
FAQs on the Greatest Common Factor of 120 and 210
1.What is the LCM of 120 and 210?
2.Is 120 divisible by 3?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 210?
5.Are 120 and 210 prime numbers?
Important Glossaries for GCF of 120 and 210
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
- Multiple: Multiples are the products we get by multiplying a given number by another. For example, the multiples of 5 are 5, 10, 15, 20, 25, 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 30 are 2, 3, and 5.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 11 is divided by 4, the remainder is 3 and the quotient is 2.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 120 and 210 is 840.
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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.
