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HCF stands for Highest Common Factor, and it is also known as the Greatest Common Divisor (GCD). LCM stands for Least Common Multiple.
What is the main difference between HCF and LCM?
HCF is the largest number that divides two or more numbers exactly, while LCM is the smallest number that is exactly divisible by the given numbers.
Can a number have more than one HCF or LCM?
No. For any set of numbers, there is only one unique HCF and one unique LCM.
What are the methods to find HCF and LCM?
You can find HCF and LCM using two main methods: the division method and prime factorization method.
Is HCF always smaller than LCM?
Yes. For any two natural numbers (except when both are equal), the HCF is always less than or equal to each number, and the LCM is always greater than or equal to each number.
HCF and LCM - Formula, Full form
HCF and LCM are used to solve problems involving division and multiples. Learn the HCF and LCM full forms and methods to find them with examples.
Nikita Aggarwal14 Jul, 2025
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The full form of HCF is the Highest Common Factor, and it is also known as the Greatest Common Divisor or GCD. The full form of LCM is Least Common Multiple. These two concepts are important in mathematics and are often used when working with numbers, especially in topics like simplification, fractions, ratios, and time-based problems.
HCF refers to the largest number that can divide two or more given numbers exactly, without leaving any remainder. On the other hand, LCM refers to the smallest number that is exactly divisible by all the given numbers.
In this blog, we will learn the steps to find both HCF and LCM using the division method and the prime factorization method using the solved example to help you understand the concept in a better way.
The LCM or Least Common Multiple of two or more numbers is the smallest number that is exactly divisible by all of the given numbers. In simple terms, it is the lowest number that appears in the multiplication tables of all the given numbers.
Example: Find the LCM of 6 and 18
Step 1: List the multiples of each number
Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Multiples of 18: 18, 36, 54, 72, ...
Step 2: Identify the common multiples
Common multiples: 18, 36, 54, ...
Step 3: Choose the lowest of these
The least common multiple is 18
Therefore, LCM of 6 and 18 is 18
HCF and LCM Formula
Let the two numbers be a and b.
There is a fundamental relationship between the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two natural numbers. This relationship states that:
HCF of a and b multiplied by LCM of a and b is equal to the product of a and b.
In mathematical terms:
HCF(a, b) × LCM(a, b) = a × b
This HCF and LCM formula is useful because if you know any two of the values (HCF, LCM, or one of the numbers), you can calculate the third.
Example
Let us take two numbers:
a = 12
b = 18
Step 1: Find the HCF of 12 and 18
The factors of 12 are: 1, 2, 3, 4, 6, 12
The factors of 18 are: 1, 2, 3, 6, 9, 18
The common factors of 12 and 18 are: 1, 2, 3, 6
So, the HCF of 12 and 18 is 6
Step 2: Use the formula to find the LCM LCM(a, b) = (a × b) ÷ HCF(a, b)
LCM(12, 18) = (12 × 18) ÷ 6 = 216 ÷ 6 = 36
Step 3: Verify the formula HCF(12, 18) × LCM(12, 18) = 6 × 36 = 216
Example: Find the HCF of 36 and 48 using the Division Method
Let us apply the steps explained above.
Step 1: Identify the smaller and larger number. Smaller number = 36 (divisor) Larger number = 48 (dividend)
Step 2: Divide 48 by 36
48 ÷ 36 = 1 remainder 12
Since the remainder is not 0, we go to the next step.
Step 3: Now use 12 (remainder) as the new divisor, and 36 (previous divisor) as the new dividend.36 ÷ 12 = 3 remainder 0
Since the remainder is now 0, we stop here.
The last divisor was 12, so the HCF of 36 and 48 is 12.
How to Find LCM by Division Method?
To find the Least Common Multiple (LCM) of two or more numbers using the division method, follow these steps:
Step 1: Start by checking if the numbers are divisible by the smallest prime number, which is 2.
Step 2: If at least one of the numbers is divisible by the chosen prime, divide it and carry the other number down as it is (if not divisible).
Step 3: Repeat the process using the same prime number until none of the numbers is divisible by it. Then move to the next prime number (3, then 5, and so on).
Step 4: Continue until all the numbers reduce to 1.
Step 5: Multiply all the prime numbers used in the divisions. The product is the LCM of the given numbers.
HCF and LCM are two important concepts in mathematics, but they serve different purposes. Understanding the difference between them helps solve problems involving division, grouping, and common multiples. Let’s explore the difference between HCF and LCM below.
Difference Between HCF and LCM
Aspects
HCF
LCM
Meaning
HCF (Highest Common Factor) is the largest number that can divide all the given numbers exactly.
LCM (Least Common Multiple) is the smallest number that all the given numbers can divide without leaving a remainder
Purpose
HCF is used when we want to split items into the largest possible equal parts.
LCM is used when we want to bring different quantities to a common multiple, like in scheduling or matching intervals.
Size Compared to the Numbers
The HCF is always less than or equal to the smallest number
The LCM is always greater than or equal to the largest number
Example
HCF of 12 and 18 = 6 (6 is the largest number that divides both 12 and 18)
LCM of 12 and 18 = 36 (36 is the smallest number that both 12 and 18 divide into)
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