What Are the Main Properties and Differences Between HCF and LCM?
The concept of Properties of HCF and LCM plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding these properties helps students solve problems related to factors, multiples, and divisibility—common topics in school exams and competitive tests.
What Is Properties of HCF and LCM?
The Highest Common Factor (HCF) is the largest number that exactly divides two or more numbers, while the Lowest Common Multiple (LCM) is the smallest number exactly divisible by two or more numbers. The properties of HCF and LCM describe unique rules and tricks for calculating, applying, and comparing them. You’ll find this concept applied in areas such as factors and multiples, time and work, and exam problem-solving.
Key Formulas for Properties of HCF and LCM
Here’s the standard formula connecting HCF and LCM for any two positive integers A and B:
HCF × LCM = Product of the numbers
\( \text{HCF}(A, B) \times \text{LCM}(A, B) = A \times B \)
Other useful formulas:
- HCF = Product of Numbers / LCM
- LCM = Product of Numbers / HCF
Key Properties of HCF
- HCF divides each number exactly: The HCF of a set of numbers exactly divides each of those numbers.
- HCF is always ≤ each individual number: The HCF cannot be larger than the smallest number in the set.
- HCF of co-prime numbers is 1: If two numbers have no common factor except 1, their HCF is 1.
- HCF is the product of common prime factors: Calculating HCF by prime factorisation involves only the common factors.
Key Properties of LCM
- LCM is always ≥ each of the numbers: The LCM can never be less than any given number.
- LCM of co-prime numbers is their product: If numbers are co-prime, LCM equals their multiplication.
- LCM is the product of the highest powers of all prime factors: When finding LCM via prime factorisation, take all primes present at maximum powers.
- Every number divides the LCM exactly: All original numbers are exact divisors of their LCM.
Product Relationship & Principle
Property: The product of the HCF and LCM of any two natural numbers is equal to the product of those two numbers.
- For A = 8 and B = 12:
HCF(8, 12) = 4, LCM(8, 12) = 24
4 × 24 = 8 × 12 = 96
This key relationship is extremely helpful for solving exams and MCQs based on properties of HCF and LCM.
When to Use HCF vs. LCM
| Use HCF | Use LCM |
|---|---|
| To split things into largest equal groups (e.g., rods, ribbons, teams) | To synchronise events (e.g., bells, traffic lights, schedule repeats) |
| To find the largest possible measurement unit | To find the earliest time or minimum quantity common to all |
Step-by-Step Illustration
Example: Find the HCF and LCM of 18 and 24 using prime factorisation and verify the relationship property.
1. Express 18 and 24 as products of prime numbers18 = 2 × 3 × 3
24 = 2 × 2 × 2 × 3
2. HCF is the product of minimum powers of all common primes:
Common primes: 2 (power 1), 3 (power 1) ⇒ HCF = 2 × 3 = 6
3. LCM is the product of maximum powers of all primes found:
LCM = 2 × 2 × 2 × 3 × 3 = 72
4. Verify property:
HCF × LCM = 6 × 72 = 432
Product of numbers = 18 × 24 = 432
5. Hence, the property holds.
Speed Trick or Vedic Shortcut
Here’s a rapid trick for co-prime numbers: The LCM of any two co-prime numbers is simply their product, and their HCF is always 1.
Example: LCM of 7 and 9 = 63 (since HCF = 1)
Such tricks reduce calculation steps in exam scenarios. Explore more Vedic Maths shortcuts in Vedantu classes!
Summarising Properties: HCF vs. LCM
| HCF | LCM |
|---|---|
| Largest common factor of numbers | Smallest common multiple of numbers |
| Always ≤ numbers given | Always ≥ numbers given |
| HCF of coprimes is 1 | LCM of coprimes is their product |
| Divides every number in the set | Divisible by every number in the set |
Try These Yourself
- Find HCF and LCM of 20 and 28, and verify their product rule.
- If the HCF of two numbers is 5 and LCM is 60, what is the product of the numbers?
- Are 14 and 25 co-prime? If yes, state their HCF and LCM.
- Split 36 mangoes and 48 oranges into largest equal groups. How many in each group?
Common Errors and Misunderstandings
- Confusing HCF with LCM in word problems.
- Forgetting to use all prime factors at correct powers in LCM.
- Assuming HCF can be greater than given numbers—never possible.
- Not checking for co-primality in shortcut tricks.
Relation to Other Concepts
The idea of properties of HCF and LCM closely relates to factors and multiples, prime factorisation, and difference between LCM and HCF. Mastering these properties helps solve a wide range of number system questions with confidence.
Classroom Tip
A quick way to remember: “HCF = Highest Common Factor, goes into numbers; LCM = Lowest Common Multiple, numbers go into it.” Vedantu’s teachers use these memory pegs and regular practice to help you score better in maths exams.
We explored Properties of HCF and LCM—including definitions, formulas, properties, sample problems, and classroom shortcuts. Practice regularly with guidance from Vedantu to master HCF and LCM for school and competitive exams!
Related Learning Resources
FAQs on Properties of HCF and LCM Explained with Examples
1. What are the rules of HCF and LCM?
The rules of HCF (Highest Common Factor) and LCM (Lowest Common Multiple) focus on understanding factors and multiples of numbers. Key principles include:
- HCF is the greatest number that divides two or more numbers without leaving a remainder.
- LCM is the smallest number that is a multiple of two or more numbers.
- For any two numbers $a$ and $b$:
$\text{HCF}(a, b)$ divides both $a$ and $b$ exactly, while $\text{LCM}(a, b)$ is divisible by both $a$ and $b$. - The product of HCF and LCM of two numbers equals the product of those numbers: $a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$.
2. What is the difference between the LCM and the HCF?
LCM (Lowest Common Multiple) is the smallest number that is a multiple of each of the numbers in question, while HCF (Highest Common Factor) is the largest number that divides each of the given numbers exactly.
- LCM focuses on multiples (e.g., for 4 and 6, LCM is 12).
- HCF focuses on common divisors (e.g., for 4 and 6, HCF is 2).
3. What is the relationship between HCF and LCM?
The relationship between HCF and LCM for any two natural numbers $a$ and $b$ is given by:
$$a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$$ This means that the product of two numbers is always equal to the product of their HCF and LCM. Vedantu simplifies this relationship with solved examples and practice exercises.
4. How to know whether to use HCF or LCM?
Use HCF when you need to divide items into the largest possible groups or to find the greatest shared factor. Use LCM when you want to calculate events happening together at the same time or to synchronize cycles. For example:
- Choose HCF for dividing something equally.
- Choose LCM for finding a common meeting time or interval.
5. What are the properties of HCF and LCM?
Some important properties of HCF and LCM include:
- Commutative property: The HCF and LCM of two numbers do not depend on their order; $\text{HCF}(a, b) = \text{HCF}(b, a)$ and $\text{LCM}(a, b) = \text{LCM}(b, a)$.
- Associative property: For three numbers, $\text{HCF}(a, (b, c)) = \text{HCF}((a, b), c)$ and $\text{LCM}(a, (b, c)) = \text{LCM}((a, b), c)$.
- Product property: $a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b)$.
- HCF of co-prime numbers: For co-prime numbers, HCF is 1 and LCM is their product.
6. How can finding the HCF and LCM be useful in daily life?
Finding the HCF and LCM helps solve real-world problems like:
- Distributing objects into groups equally (using HCF).
- Synchronizing events, such as ringing bells or traffic lights (using LCM).
- Figuring out the most efficient ways to share things or set repeating schedules.
7. What are the common methods to find HCF and LCM of numbers?
To find the HCF, common methods include:
- Prime factorization method: Write each number as a product of prime numbers and multiply the common prime factors.
- Division method: Repeatedly divide the larger number by the smaller.
- Prime factorization method: Multiply each prime factor the greatest number of times it appears in any of the numbers.
- Listing multiples: List multiples and find the smallest one common to all numbers.
8. Can HCF and LCM be found for more than two numbers?
Yes, you can find the HCF and LCM for more than two numbers. For HCF, find the highest factor common to all the numbers. For LCM, find the smallest number that all given numbers divide into evenly. Advanced examples and strategies are provided by Vedantu to help students tackle complex problems involving three or more numbers.
9. What is the HCF and LCM of co-prime numbers?
For co-prime numbers (numbers whose only common factor is 1):
- HCF is always 1.
- LCM is equal to the product of the two numbers. For numbers $a$ and $b$, if $\gcd(a, b) = 1$, then: $\text{LCM}(a, b) = a \times b$
10. How does understanding HCF and LCM help in competitive exams?
A strong understanding of HCF and LCM properties helps students:
- Solve quantitative aptitude questions quickly.
- Handle word problems involving groups, intervals, or distributions.
- Score better on examinations like the Olympiad, NTSE, and various school entrance exams.
