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107 LearnersLast updated on August 12, 2025
GCF of 14 and 84
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 schedule events. In this topic, we will learn about the GCF of 14 and 84.
What is the GCF of 14 and 84?
The greatest common factor of 14 and 84 is 14. 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 14 and 84?
To find the GCF of 14 and 84, a few methods are described below -
- Listing Factors
- Prime Factorization
- Long Division Method / by Euclidean Algorithm
GCF of 14 and 84 by Using Listing of Factors
Steps to find the GCF of 14 and 84 using the listing of factors:
Step 1: Firstly, list the factors of each number
Factors of 14 = 1, 2, 7, 14.
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 14 and 84: 1, 2, 7, 14.
Step 3: Choose the largest factor The largest factor that both numbers have is 14. The GCF of 14 and 84 is 14.
GCF of 14 and 84 Using Prime Factorization
To find the GCF of 14 and 84 using the Prime Factorization Method, follow these steps:
Step 1: Find the prime factors of each number
Prime Factors of 14: 14 = 2 × 7
Prime Factors of 84: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7
Step 2: Now, identify the common prime factors The common prime factors are: 2 × 7
Step 3: Multiply the common prime factors The GCF is 2 × 7 = 14.
The Greatest Common Factor of 14 and 84 is 14.
GCF of 14 and 84 Using Division Method or Euclidean Algorithm Method
Find the GCF of 14 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 14 84 ÷ 14 = 6 (quotient), The remainder is calculated as 84 − (14×6) = 0
The remainder is zero, so the divisor will become the GCF. The GCF of 14 and 84 is 14.
Common Mistakes and How to Avoid Them in GCF of 14 and 84
Finding the GCF of 14 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 14, 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 indicates an 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 14 and 84 Examples
Problem 1
A gardener has 14 roses and 84 tulips. She wants to arrange them into bouquets with the largest number of flowers in each bouquet. How many flowers will be in each bouquet?
We should find the GCF of 14 and 84 GCF of 14 and 84 is 14.
There are 14 flowers in each bouquet. 14 ÷ 14 = 1 84 ÷ 14 = 6
There will be 7 bouquets, and each bouquet gets 1 rose and 6 tulips.
Explanation
As the GCF of 14 and 84 is 14, the gardener can make 7 bouquets. Now divide 14 and 84 by 14. Each bouquet gets 1 rose and 6 tulips.
Problem 2
A school has 14 small tables and 84 chairs. They want to arrange them in sets with the same number of tables and chairs in each set, using the largest possible number of items per set. How many items will be in each set?
GCF of 14 and 84 is 14. So each set will have 14 items.
Explanation
There are 14 tables and 84 chairs. To find the total number of items in each set, we should find the GCF of 14 and 84. There will be 14 items in each set.
Problem 3
A chef has 14 kg of rice and 84 kg of wheat. She wants to pack them into bags of equal weight, using the largest possible weight for each bag. What should be the weight of each bag?
For calculating the largest equal weight, we have to calculate the GCF of 14 and 84. The GCF of 14 and 84 is 14. The weight of each bag is 14 kg.
Explanation
For calculating the largest weight of the bags, we first need to calculate the GCF of 14 and 84, which is 14. The weight of each bag will be 14 kg.
Problem 4
A carpenter has two wooden planks, one 14 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 14 and 84 is 14. The longest length of each piece is 14 cm.
Explanation
To find the longest length of each piece of the two wooden planks, 14 cm and 84 cm, respectively, we have to find the GCF of 14 and 84, which is 14 cm. The longest length of each piece is 14 cm.
Problem 5
If the GCF of 14 and ‘b’ is 14, and the LCM is 168, find ‘b’.
The value of ‘b’ is 84.
Explanation
GCF × LCM = product of the numbers
14 × 168 = 14 × b
2352 = 14b
b = 2352 ÷ 14 = 168
FAQs on the Greatest Common Factor of 14 and 84
1.What is the LCM of 14 and 84?
2.Is 14 divisible by 2?
3.What will be the GCF of any two prime numbers?
4.What is the prime factorization of 84?
5.Are 14 and 84 prime numbers?
Important Glossaries for GCF of 14 and 84
- Factors: Factors are numbers that divide the target number completely. For example, the factors of 14 are 1, 2, 7, and 14.
- Multiple: Multiples are the products we get by multiplying a given number by another. For example, the multiples of 7 are 7, 14, 21, 28, 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 14 are 2 and 7.
- Remainder: The value left after division when the number cannot be divided evenly. For example, when 84 is divided by 14, the remainder is 0.
- LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 14 and 84 is 84.
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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.
