368 LearnersLast updated on June 4th, 2025
HCF (GCF)
When two or more non-zero integers, let’s say x and y have common factors a, b, c, and d in ascending order. Then the common factor d is the highest common factor (HCF) for both x and y.
What is HCF in Math?
The Highest Common Factor (HCF) also known as the Greatest Common Factor (GCF) is used in the field of mathematics to simply fractions and understand ratios. By dividing the given numbers by their HCF give an equivalent fraction in their simplest form.
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History and Origin of HCF
This mathematical concept can be traced back to ancient Babylonian mathematics (circa 1800 BCE). The Babylonians used HCF algorithms to solve problems involving numbers with fractions and divisors in their calculations. The use of HCF was then formalized by a Greek mathematician named Euclid in his book Elements.
Properties of HCF
The highest number that divides exactly two or more numbers is called the Highest Common Factor (HCF) or the Greatest Common Factor (GCF). HCF is usually studied along with its counterpart known as the Least Common Multiple (LCM), or the Lowest Common Multiple (LCM). LCM is the smallest common multiple between two or more numbers that can divide the given numbers without leaving any remainder. The formula linking both HCF and LCM is:
HCF (a, b) LCM (a, b) = a b
This helps determine the HCF when the LCM is known and vice versa and highlights the inverse relationship between HCF and LCM in number theory. The HCF can be calculated using three different methods:
- Listing out common factors
- Prime factorization
- Division method.
Methods to Find HCF
The HCF (Highest Common Factor) can be found using methods like Prime Factorization, which identifies and multiplies common prime factors, Division, which repeatedly divides numbers until the remainder is zero, and Listing Factors, which compares all factors to find the greatest one. Here we have given three different methods to find the HCF of numbers:
Prime Factorization Method
Prime factorization is the process of expressing a number as the product of its prime factors, which are the smallest prime numbers that multiply together to equal the original number. To find the HCF of the given numbers, we need to find the common prime factors
For example, to find the HCF of 200 and 300, prime factorize them both:
Prime Factorization of 200 = 23 52
Prime Factorization of 300 = 22 3 52
The common factors among both are: 2, 2, 5, 5
The HCF of 200 and 300 = 2 2 5 5 = 100
Thus, the HCF is 100.
Division Method
The division method for finding the HCF involves dividing the larger number by the smaller number, then using the remainder as the new divisor, and repeating the process until the remainder becomes zero. The last non-zero divisor is the HCF.
For example, to find the HCF of 200 and 300, divide the larger number by the smaller number:
That is, 300 200 = 1 (quotient) and the remainder as 100.
Next, divide replacing the larger number (300) with the smaller number (200), and the smaller number (200) with the remainder (100), repeating until the remainder is 0, where the last non-zero number is the HCF.
That is, divide 200 by 100:
200 100 = 2 (quotient) and the remainder 0.
Since we got 0 as the remainder, 100 is the HCF.
Thus, the HCF of 200 and 300 is 100.
Listing Factors Method
In this method, the factors of both the numbers are listed, thereby manually checking the common factors among them and then finding the highest common factor from the lot.
For example, let’s take 200 and 300:
Factors of 200 = 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, and 200.
Factors of 300 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, and 300.
Common Factors of 200 and 300 = 1, 2, 4, 5, 10, 20, 25, 50, 100
The highest common factor from the common factors = 100
Hence, the HCF of 200 and 300 is 100.
Tips and Tricks to Master HCF
While finding the HCF of numbers, using these tips and tricks will help you reach the answers faster. Practice identifying these patterns and shortcuts below.
- If there are more than two numbers to find the HCF, first find the HCF between two numbers and then find the HCF of the third number using the result of the first two numbers.
- The HCF of co-prime numbers are always 1. Co-prime numbers are numbers that have no common factors other than 1. Finding HCF among prime numbers is far easier and faster.
- If one number is a factor of another number, then the smaller number is the HCF of both numbers.
Common Mistakes and How to Avoid Them in HCF
There is always a possibility that the kids might make typical errors while finding the Highest Common Factor (HCF) of given numbers. Here we have given some key mistakes and how to prevent those errors.
Mistake 1
Incorrect Prime Factorization
Kids might incorrectly factorize the numbers into the prime factors. For example, while finding the HCF of 200 and 300, they might incorrectly write 200 as 22 52 instead of 23 52. Always double-check the prime factorization process and multiply the factorized numbers again to get the original value.
Mistake 2
Confusing HCF with LCM
Kids might mix up the two concepts HCF and LCM. For example, they might mistakenly calculate LCM for HCF and HCF for LCM. Make sure that the kids understand the full forms of HCF and LCM and how to find both separately.
Mistake 3
Skipping Common Factors
When looking for the common factors of two or more numbers, children may skip a number. For instance, they might omit 100 when determining the HCF of 200 and 300 and instead determine that 50 is the HCF of 200 and 300. Make sure children always write down each number individually before identifying the common ones without omitting any.
Mistake 4
Not Simplifying Word Problems
In word problems, kids might fail to understand whether they should find HCF or LCM. As both of these are learned parallelly, they might be confused between both. Ask them to carefully read and analyze the problem.
Mistake 5
Ignoring Special Cases
Special cases like co-prime numbers have HCF as 1 will be mishandled by kids. They will try hard to figure out the prime factors of prime numbers without thinking that prime numbers don’t have factors other than 1 and itself. Teach kids that prime numbers will always have HCF as 1.
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Solved Examples on GCF
Problem 1
Imagine you have 36 and 48 candies and want to divide them into the largest equal-sized bags without any leftovers. How many candies will be in each bag?
There will be 12 candies per bag.
Explanation
First, find the factors of 36 and 48
Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
The common factors among them both = 1, 2, 3, 4, 6, 12
The highest common factor = 12
Thus, each bag can hold 12 candies without any leftovers.
Problem 2
You have two ropes, one 72 cm long and the other 96 cm long. You want to cut them into pieces of equal length, with no leftover ropes. What is the maximum length of each piece?
The maximum length of each rope will be 24 cm.
Explanation
First, find the factors of 72 and 96
Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 96 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
The common factors among them both = 1, 2, 3, 4, 6, 8, 12, 24
The highest common factor = 24
Thus, each rope piece can be 24 cm long.
Problem 3
A teacher has 60 pencils and 84 erasers. She wants to distribute them equally to students in sets, with no leftovers. How many sets can she make, and how many items will be in each set?
There will be 12 sets, each with 12 items.
Explanation
First, find the factors of 60 and 84
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
The common factors among them both = 1, 2, 3, 4, 6, 12
The highest common factor = 12
Thus, she can create 12 sets with 12 items in each set.
Problem 4
Two people work out on different days of the week, one on an 18-day cycle and the other on a 24-day cycle. How many days will it be before they both work out on the same day again?
There will be 72 days before they both work out on the same day again.
Explanation
First, find the factors of 18 and 24
Factors of 18 = 1, 2, 3, 6, 9, 18
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
The common factors among them both = 1, 2, 3, 4, 6
The highest common factor = 6
Now, find the least common multiple (LCM) of 18 and 24
The LCM of 18 and 24 = (18 24) 6 = 72
Hence, the two people will both work out on the same day again after 72 days.
Problem 5
You are tiling a floor that measures 120 cm by 180 cm using square tiles. What is the largest possible size of the tile that will fit perfectly into the floor without cutting any tiles?
The largest possible tile size is 60 cm.
Explanation
First, find the factors of 120 and 180
Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Factors of 180 = 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
The common factors among them both = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
The highest common factor = 60
Thus, the largest tile size will be 60 cm by 60 cm to fit perfectly.
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FAQs on Highest Common Factor (HCF)
1.What is HCF?
2.How is HCF calculated?
3.What is the HCF of two numbers?
4.How can the HCF of three digits be found?
5.How do HCF and LCM differ from one another?
6.Is it always true that the HCF of two numbers is a factor of both?
7.What is the HCF of prime numbers?
8.What occurs if two numbers have an HCF of 1?
9.Is it possible for the HCF of two numbers to exceed both of them?
10.What is the practical application of HCF?
11.How can children in India use numbers in everyday life to understand HCF (GCF) ?
12.What are some fun ways kids in India can practice HCF (GCF) with numbers?
13.What role do numbers and HCF (GCF) play in helping children in India develop problem-solving skills?
14.How can families in India create number-rich environments to improve HCF (GCF) skills?
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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.
