Mastering LCM and HCF for Competitive Exams
Welcome to the foundational module on Least Common Multiple (LCM) and Highest Common Factor (HCF), also known as Greatest Common Divisor (GCD). These concepts are fundamental to many quantitative aptitude questions in competitive exams like the CAT. Understanding them thoroughly will unlock your ability to solve a wide range of problems efficiently.
What is HCF (Highest Common Factor)?
The HCF of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. It's also known as the Greatest Common Divisor (GCD).
HCF is the largest number that divides all given numbers exactly.
Think of HCF as the biggest 'common building block' that can be used to construct all the given numbers through multiplication. For example, the HCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 (as 6 x 2) and 18 (as 6 x 3) without a remainder.
To find the HCF, we can list all the factors of each number and identify the largest factor that appears in all lists. Alternatively, prime factorization is a more systematic method. We find the prime factorization of each number and then multiply the common prime factors raised to the lowest power they appear in any of the factorizations.
The HCF of 24 and 36 is 12. (Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. The highest common factor is 12.)
What is LCM (Least Common Multiple)?
The LCM of two or more numbers is the smallest positive integer that is a multiple of all the numbers. It's the smallest number that all the given numbers can divide into evenly.
LCM is the smallest number that all given numbers can divide into.
Imagine listing out the multiples of each number. The LCM is the first number that appears in all of these lists. For instance, the LCM of 4 and 6 is 12. Multiples of 4: 4, 8, 12, 16, 20, 24... Multiples of 6: 6, 12, 18, 24... The smallest common multiple is 12.
The most efficient way to find the LCM is by using prime factorization. We find the prime factorization of each number and then multiply all the prime factors (from all numbers) raised to the highest power they appear in any of the factorizations. Another method is to use the formula: LCM(a, b) = (a * b) / HCF(a, b).
The LCM of 8 and 12 is 24. (Multiples of 8: 8, 16, 24, 32... Multiples of 12: 12, 24, 36... The smallest common multiple is 24.)
Relationship Between HCF and LCM
There's a crucial relationship between the HCF and LCM of two numbers, which is often tested in competitive exams. For any two positive integers 'a' and 'b':
<b>HCF(a, b) × LCM(a, b) = a × b</b>
This formula is a powerful shortcut. If you know any three of the values (a, b, HCF, or LCM), you can easily find the fourth.
Let's visualize the prime factorization method for finding HCF and LCM. Consider the numbers 12 and 18.
Prime factorization of 12: 2 x 2 x 3 = 2² x 3¹ Prime factorization of 18: 2 x 3 x 3 = 2¹ x 3²
For HCF, we take the common prime factors raised to their lowest power: HCF(12, 18) = 2¹ x 3¹ = 6.
For LCM, we take all prime factors raised to their highest power: LCM(12, 18) = 2² x 3² = 4 x 9 = 36.
Let's check the relationship: HCF x LCM = 6 x 36 = 216. And a x b = 12 x 18 = 216. The relationship holds!
Text-based content
Library pages focus on text content
Methods for Calculating HCF and LCM
| Method | Description | Best For |
|---|---|---|
| Listing Factors/Multiples | Write out all factors or multiples and find the common ones. | Small numbers, conceptual understanding |
| Prime Factorization | Break down numbers into their prime factors. | Most numbers, systematic approach |
| Division Method (for HCF) | Repeatedly divide the numbers by common prime factors. | Multiple numbers, efficient calculation |
| Division Method (for LCM) | Divide numbers by prime factors, bringing down non-divisible numbers. | Multiple numbers, efficient calculation |
| Formula: LCM = (a*b)/HCF | Use the relationship between HCF and LCM. | When HCF is known or easily found, or when dealing with two numbers |
Practice Problems & Strategies
When solving problems involving LCM and HCF, always identify what the question is asking for: the largest common factor or the smallest common multiple. Pay close attention to keywords like 'greatest', 'largest', 'smallest', 'least', 'simultaneously', 'together', 'first time again'. Understanding the context of the problem will guide you to the correct concept and method.
Using the formula HCF x LCM = Product of numbers, we get 12 x LCM = 1080. Therefore, LCM = 1080 / 12 = 90.
Learning Resources
A comprehensive explanation of HCF and LCM concepts, including various methods and formulas, with practice examples.
This resource provides clear definitions, methods, and properties of LCM and HCF, suitable for building a strong foundation.
Offers a collection of practice questions with solutions to reinforce understanding and test application of HCF and LCM concepts.
Explains the core concepts of HCF and LCM with illustrative examples and practical applications.
Focuses on HCF and LCM specifically for CAT aspirants, covering important formulas and question types.
A visual tutorial demonstrating how to find HCF and LCM using the prime factorization method.
A concise summary of all essential formulas and properties related to HCF and LCM for quick revision.
Khan Academy's introduction to divisors and multiples, covering GCD (HCF) with clear explanations.
Explains LCM and provides examples, including word problems that require understanding of LCM and HCF.
A discussion and explanation of HCF and LCM within the context of CAT preparation, offering insights and tips.