Have you ever needed to split two different groups of items into the largest possible equal sizes? Whether you are organizing a classroom activity or solving a math homework problem, understanding the Highest Common Factor (H.C.F) is the key.
In this post, we’ll break down exactly what H.C.F is and show you the easiest way to find it.
The Highest Common Factor (H.C.F)—also known as the Greatest Common Divisor (G.C.D)—is the largest number that divides two or more numbers exactly without leaving a remainder.
Definition: The greatest number which is a common factor of two (or more) given numbers is called the Highest Common Factor.
One of the most straightforward ways to find the H.C.F is by listing out all the factors of each number and identifying the largest one they share.
Let’s walk through the solution step-by-step:
Step 1: List the factors of the first number
Factors of 6: 1, 2, 3, 6
Step 2: List the factors of the second number
Factors of 8: 1, 2, 4, 8
Step 3: Identify the common factors
Look at both lists and find the numbers that appear in both:
Common factors: 1, 2
Step 4: Choose the highest number
Out of the common factors (1 and 2), the largest number is 2.
Final Answer:
H.C.F = 2
Mastering H.C.F is a foundational skill in mathematics. It helps you:
Simplify fractions to their lowest terms.
Solve real-world “distribution” problems.
Prepare for more complex topics like Algebra and Ratio.
| Number | Factors | Common Factors | H.C.F |
| 6 | 1, 2, 3, 6 | 1, 2 | 2 |
| 8 | 1, 2, 4, 8 | 1, 2 | 2 |
Building on our previous guide, let’s explore more advanced ways to find the Highest Common Factor (H.C.F). While listing factors is great for small numbers, these two methods are much faster for larger calculations.
This method involves breaking each number down into its “prime building blocks.”
Prime Factorize: Divide each number by prime numbers (2, 3, 5, etc.) until you reach 1.
Identify Common Bases: Look for the prime factors that appear in both numbers.
Multiply Common Factors: Multiply these common prime factors together to get your H.C.F.
Example: H.C.F of 6 and 8
$6 = 2^1 \times 3^1$
$8 = 2^3 \times 3^0$
The common factor here is $2^1$.
H.C.F = 2
The Division Method is often the quickest way to find the H.C.F of very large numbers.
Divide: Divide the larger number by the smaller number.
Use the Remainder: Take the remainder from the first step and make it the new divisor.
Repeat: Divide the previous divisor by this new remainder.
Finish: Continue this process until the remainder is zero.
Result: The last divisor used is your H.C.F.
Example: H.C.F of 6 and 8
$8 \div 6$ gives a remainder of 2.
Now, divide the previous divisor (6) by the remainder (2): $6 \div 2 = 3$ with a remainder of 0.
Since the last divisor used was 2, the H.C.F is 2.
You can use these same methods for more than two numbers. Let’s look at the H.C.F of 24, 36, and 84 using the “Listing Factors” method:
Factors of 24: 1, 2, 3, 4, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 26, 42, 84
Common Factors: 1, 2, 3, 4, 12
Highest Common Factor: 12
Pro Tip: If you have more than two numbers and want to use the Division Method, simply find the H.C.F of the first two numbers, then find the H.C.F of that result and the third number!
In our previous guide, we looked at listing factors to find the Highest Common Factor (H.C.F). While that works for small numbers, it can get tricky as numbers grow. Here are two more powerful methods to master H.C.F like a pro!
This method involves breaking numbers down into their simplest “prime” components.
The Process:
Prime factorize each given number.
Identify the common factors.
Multiply these common factors together to get the H.C.F.
Example: H.C.F of 24, 36, and 84
$24 = 2^3 \times 3^1 \times 7^0$
$36 = 2^2 \times 3^2 \times 7^0$
$84 = 2^2 \times 3^1 \times 7^1$
Common Factors: $2^2 \times 3^1 \times 7^0$
H.C.F: $4 \times 3 = \mathbf{12}$
This is often the fastest way to handle large numbers.
The Process:
Divide the greater number by the smaller number.
Take the remainder and make it your new divisor.
Divide the previous divisor by this new remainder.
Repeat until you get a remainder of zero.
The last divisor used is the H.C.F.
Example: H.C.F of 24, 36, and 84
First, find the H.C.F of 24 and 84:
$84 \div 24$ leaves a remainder of 12.
Now divide the previous divisor (24) by the remainder (12): $24 \div 12 = 2$ with 0 remainder.
The H.C.F of 24 and 84 is 12.
Next, check 12 against the remaining number, 36:
$36 \div 12 = 3$ with 0 remainder.
Final H.C.F = 12.
If you are given numbers already in exponent form, like $2^{19} \times 3^{82}$ and $2^{51} \times 3^{54}$, the H.C.F is simply the product of the lowest powers of the common prime bases.
Two numbers are called Relative Prime or Co-primes if they have no common factors other than 1.
The Rule: $a, b$ are co-primes if their G.C.D (H.C.F) is 1.
Try finding the H.C.F for these sets using your favorite method:
52 and 48
15 and 35
36, 48, and 52
60, 128, and 180