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102 LearnersLast updated on September 24, 2025
GCF of 84 and 90
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 84 and 90.
What is the GCF of 84 and 90?
The greatest common factor of 84 and 90 is 6. 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 84 and 90?
To find the GCF of 84 and 90, a few methods are described below
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 84 and 90 by Using Listing of Factors
Steps to find the GCF of 84 and 90 using the listing of factors:
Step 1: Firstly, list the factors of each number
Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
Factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
Step 2: Now, identify the common factors of them Common factors of 84 and 90: 1, 2, 3, 6.
Step 3: Choose the largest factor The largest factor that both numbers have is 6.
The GCF of 84 and 90 is 6.
GCF of 84 and 90 Using Prime Factorization
To find the GCF of 84 and 90 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number
Prime Factors of 84: 84 = 2 x 2 x 3 x 7 = 2² x 3 x 7
Prime Factors of 90: 90 = 2 x 3 x 3 x 5 = 2 x 3² x 5
Step 2: Now, identify the common prime factors The common prime factors are: 2 x 3 = 2 x 3
Step 3: Multiply the common prime factors 2 x 3 = 6.
The Greatest Common Factor of 84 and 90 is 6.
GCF of 84 and 90 Using Division Method or Euclidean Algorithm Method
Find the GCF of 84 and 90 using the division method or Euclidean Algorithm Method. Follow these steps:
Step 1: First, divide the larger number by the smaller number
Here, divide 90 by 84 90 ÷ 84 = 1 (quotient),
The remainder is calculated as 90 − (84×1) = 6
The remainder is 6, not zero, so continue the process
Step 2: Now divide the previous divisor (84) by the previous remainder (6)
Divide 84 by 6 84 ÷ 6 = 14 (quotient), remainder = 84 − (6×14) = 0
The remainder is zero, the divisor will become the GCF.
The GCF of 84 and 90 is 6.
Common Mistakes and How to Avoid Them in GCF of 84 and 90
Finding GCF of 84 and 90 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 84, students may mention 5 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 that 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 84 and 90 Examples
Problem 1
A gardener has 84 tulips and 90 roses. She wants to plant them in equal groups with the largest number of flowers in each group. How many flowers will be in each group?
We should find the GCF of 84 and 90 GCF of 84 and 90 2 x 3 = 6.
There are 6 equal groups 84 ÷ 6 = 14 90 ÷ 6 = 15
There will be 6 groups, and each group gets 14 tulips and 15 roses.
Explanation
As the GCF of 84 and 90 is 6, the gardener can make 6 groups.
Now divide 84 and 90 by 6.
Each group gets 14 tulips and 15 roses.
Problem 2
A school has 84 red chairs and 90 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 84 and 90 2 x 3 = 6.
So each row will have 6 chairs.
Explanation
There are 84 red and 90 blue chairs.
To find the total number of chairs in each row, we should find the GCF of 84 and 90.
There will be 6 chairs in each row.
Problem 3
A tailor has 84 meters of red ribbon and 90 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 84 and 90
The GCF of 84 and 90 2 x 3 = 6.
The ribbon is 6 meters long.
Explanation
For calculating the longest length of the ribbon first we need to calculate the GCF of 84 and 90 which is 6.
The length of each piece of the ribbon will be 6 meters.
Problem 4
A carpenter has two wooden planks, one 84 cm long and the other 90 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 84 and 90 2 x 3 = 6.
The longest length of each piece is 6 cm.
Explanation
To find the longest length of each piece from the two wooden planks, 84 cm and 90 cm, respectively.
We have to find the GCF of 84 and 90, which is 6 cm.
The longest length of each piece is 6 cm.
Problem 5
If the GCF of 84 and ‘b’ is 6, and their LCM is 1260. Find ‘b’.
The value of ‘b’ is 90.
Explanation
GCF x LCM = product of the numbers
6 × 1260 = 84 × b
7560 = 84b
b = 7560 ÷ 84 = 90
FAQs on the Greatest Common Factor of 84 and 90
1.What is the LCM of 84 and 90?
2.Is 84 divisible by 2?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 90?
5.Are 84 and 90 prime numbers?
Important Glossaries for GCF of 84 and 90
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 18 are 1, 2, 3, 6, 9, and 18.
- 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 20 are 2 and 5.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 13 is divided by 5, 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 5 and 6 is 30.
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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.
