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101 LearnersLast updated on September 9, 2025
GCF of 60 and 84
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 schedule events. In this topic, we will learn about the GCF of 60 and 84.
What is the GCF of 60 and 84?
The greatest common factor of 60 and 84 is 12. 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 60 and 84?
To find the GCF of 60 and 84, a few methods are described below
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 60 and 84 by Using Listing of Factors
Steps to find the GCF of 60 and 84 using the listing of factors
Step 1: Firstly, list the factors of each number
Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
Step 2: Now, identify the common factors of them Common factors of 60 and 84: 1, 2, 3, 4, 6, 12.
Step 3: Choose the largest factor The largest factor that both numbers have is 12. The GCF of 60 and 84 is 12.
GCF of 60 and 84 Using Prime Factorization
To find the GCF of 60 and 84 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number Prime Factors of 60: 60 = 2 x 2 x 3 x 5 = 2² x 3 x 5 Prime Factors of 84: 84 = 2 x 2 x 3 x 7 = 2² x 3 x 7
Step 2: Now, identify the common prime factors The common prime factors are: 2 x 2 x 3 = 2² x 3
Step 3: Multiply the common prime factors 2² x 3 = 4 x 3 = 12. The Greatest Common Factor of 60 and 84 is 12.
GCF of 60 and 84 Using Division Method or Euclidean Algorithm Method
Find the GCF of 60 and 84 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number Here, divide 84 by 60 84 ÷ 60 = 1 (quotient), remainder is calculated as 84 − (60×1) = 24 The remainder is 24, not zero, so continue the process
Step 2: Now divide the previous divisor (60) by the previous remainder (24) Divide 60 by 24 60 ÷ 24 = 2 (quotient), remainder = 60 − (24×2) = 12 The remainder is 12, not zero, so continue the process
Step 3: Now divide the previous divisor (24) by the previous remainder (12) Divide 24 by 12 24 ÷ 12 = 2 (quotient), remainder = 24 − (12×2) = 0 The remainder is zero, the divisor will become the GCF. The GCF of 60 and 84 is 12.
Common Mistakes and How to Avoid Them in GCF of 60 and 84
Finding GCF of 60 and 84 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 60, students may mention 50, 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 tells about the 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 60 and 84 Examples
Problem 1
A chef has 60 apples and 84 oranges. He 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 60 and 84 GCF of 60 and 84 2² x 3 = 4 x 3 = 12. There are 12 equal groups 60 ÷ 12 = 5 84 ÷ 12 = 7 There will be 12 groups, and each group gets 5 apples and 7 oranges.
Explanation
As the GCF of 60 and 84 is 12, the chef can make 12 groups.
Now divide 60 and 84 by 12.
Each group gets 5 apples and 7 oranges.
Problem 2
A school has 60 red chairs and 84 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 60 and 84 2² x 3 = 4 × 3 = 12. So each row will have 12 chairs.
Explanation
There are 60 red and 84 blue chairs.
To find the total number of chairs in each row, we should find the GCF of 60 and 84.
There will be 12 chairs in each row.
Problem 3
A tailor has 60 meters of red ribbon and 84 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 60 and 84 The GCF of 60 and 84 2² x 3 = 4 × 3 = 12. The ribbon is 12 meters long.
Explanation
For calculating the longest length of the ribbon, first we need to calculate the GCF of 60 and 84, which is 12.
The length of each piece of the ribbon will be 12 meters.
Problem 4
A carpenter has two wooden planks, one 60 cm long and the other 84 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 60 and 84 2² x 3 = 4 × 3 = 12. The longest length of each piece is 12 cm.
Explanation
To find the longest length of each piece of the two wooden planks, 60 cm and 84 cm respectively, we have to find the GCF of 60 and 84, which is 12 cm.
The longest length of each piece is 12 cm.
Problem 5
If the GCF of 60 and ‘a’ is 12, and the LCM is 420, find ‘a’.
The value of ‘a’ is 84.
Explanation
GCF x LCM = product of the numbers
12 × 420
= 60 × a 5040
= 60a a
= 5040 ÷ 60 = 84
FAQs on the Greatest Common Factor of 60 and 84
1.What is the LCM of 60 and 84?
2.Is 60 divisible by 2?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 84?
5.Are 60 and 84 prime numbers?
Important Glossaries for GCF of 60 and 84
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
- Multiple: Multiples are the products we get by multiplying a given number by another. For example, the multiples of 4 are 4, 8, 12, 16, 20, 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 15 are 3 and 5.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 12 is divided by 7, the remainder is 5 and the quotient is 1.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 60 and 84 is 420.
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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.
