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102 LearnersLast updated on September 23, 2025
GCF of 72 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 72 and 84.
What is the GCF of 72 and 84?
The greatest common factor of 72 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 72 and 84?
To find the GCF of 72 and 84, a few methods are described below -
Listing Factors Prime Factorization Long Division Method / by Euclidean Algorithm
GCF of 72 and 84 by Using Listing of Factors
Steps to find the GCF of 72 and 84 using the listing of factors
Step 1: Firstly, list the factors of each number Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. 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 72 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 72 and 84 is 12.
GCF of 72 and 84 Using Prime Factorization
To find the GCF of 72 and 84 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number Prime Factors of 72: 72 = 2 x 2 x 2 x 3 x 3 = 2^3 x 3^2 Prime Factors of 84: 84 = 2 x 2 x 3 x 7 = 2^2 x 3 x 7.
Step 2: Now, identify the common prime factors The common prime factors are: 2 x 2 x 3 = 2^2 x 3.
Step 3: Multiply the common prime factors 2^2 x 3 = 4 × 3 = 12.
The Greatest Common Factor of 72 and 84 is 12.
GCF of 72 and 84 Using Division Method or Euclidean Algorithm Method
Find the GCF of 72 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 72 84 ÷ 72 = 1 (quotient), The remainder is calculated as 84 − (72×1) = 12 The remainder is 12, not zero, so continue the process.
Step 2: Now divide the previous divisor (72) by the previous remainder (12) Divide 72 by 12 72 ÷ 12 = 6 (quotient), remainder = 72 − (12×6) = 0 The remainder is zero, the divisor will become the GCF.
The GCF of 72 and 84 is 12.
Common Mistakes and How to Avoid Them in GCF of 72 and 84
Finding GCF of 72 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 72, 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 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 72 and 84 Examples
Problem 1
A chef has 72 apples and 84 oranges. She wants to distribute them equally among her staff, with the largest possible number of fruits in each group. How many fruits will be in each group?
We should find the GCF of 72 and 84 GCF of 72 and 84 2^2 x 3 = 4 x 3 = 12.
There are 12 equal groups 72 ÷ 12 = 6 84 ÷ 12 = 7.
There will be 12 groups, and each group gets 6 apples and 7 oranges.
Explanation
As the GCF of 72 and 84 is 12, the chef can make 12 groups.
Now divide 72 and 84 by 12.
Each group gets 6 apples and 7 oranges.
Problem 2
A gardener has 72 rose bushes and 84 tulip bulbs. They want to arrange them in rows with the same number of plants in each row, using the largest possible number of plants per row. How many plants will be in each row?
GCF of 72 and 84 2^2 x 3 = 4 × 3 = 12.
So each row will have 12 plants.
Explanation
There are 72 rose bushes and 84 tulip bulbs.
To find the total number of plants in each row, we should find the GCF of 72 and 84.
There will be 12 plants in each row.
Problem 3
A tailor has 72 meters of silk and 84 meters of cotton 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 72 and 84 The GCF of 72 and 84 2^2 x 3 = 4 × 3 = 12.
The fabric is 12 meters long.
Explanation
For calculating the longest length of the fabric, first, we need to calculate the GCF of 72 and 84, which is 12.
The length of each piece of fabric will be 12 meters.
Problem 4
A carpenter has two wooden planks, one 72 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 72 and 84 2^2 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, 72 cm and 84 cm, respectively, we have to find the GCF of 72 and 84, which is 12 cm.
The longest length of each piece is 12 cm.
Problem 5
If the GCF of 72 and ‘b’ is 12, and the LCM is 504. Find ‘b’.
The value of ‘b’ is 84.
Explanation
GCF x LCM = product of the numbers 12 × 504 = 72 × b 6048 = 72b b = 6048 ÷ 72 = 84
FAQs on the Greatest Common Factor of 72 and 84
1.What is the LCM of 72 and 84?
2.Is 72 divisible by 2?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 84?
5.Are 72 and 84 prime numbers?
Important Glossaries for GCF of 72 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.
- 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 72 and 84 is 504.
- GCF: The largest factor that commonly divides two or more numbers. For example, the GCF of 72 and 84 is 12, 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.
