Powered by (gatecareer.com)
LCM & HCF
Master LCM and HCF with simple explanations, methods, and examples. Perfect for competitive exams and real-life applications
When studying for competitive exams or improving mathematical aptitude, two crucial concepts that often arise are LCM (Least Common Multiple) and HCF (Highest Common Factor). They may sound complicated initially, but with the right approach and understanding, they become much easier to comprehend and apply in various problems. This blog will explore these concepts, break them down into simple language, and provide detailed examples to help you master the topic.
LCM
What is LCM?
The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers. In other words, it’s the smallest number that all the given numbers divide without leaving a remainder.
Example: Let’s take two numbers, 4 and 6.
- The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, …
- The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, …
The smallest multiple common to both lists is 12. Hence, LCM (4, 6) = 12.
What is HCF?
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers without leaving a remainder. It is the greatest factor common to all the numbers involved.
Example: Let’s take two numbers, 24 and 36.
- The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
- The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.
The greatest number common to both lists of factors is 12. Hence, HCF (24, 36) = 12.
HCF
Section 1
Methods to Find LCM and HCF
There are several methods to find the LCM and HCF of given numbers. Below, we will explore the most common methods.
Prime Factorization Method
The Prime Factorization method involves breaking down the numbers into their prime factors and then finding the LCM and HCF from those factors.
Steps to Find LCM:
- Write the prime factorization of each number.
- Take the highest power of each prime number.
- Multiply the highest powers to get the LCM.
Steps to Find HCF:
- Write the prime factorization of each number.
- Take the lowest power of each common prime factor.
- Multiply the lowest powers to get the HCF.
Example:
Let’s take two numbers, 18 and 24.
- Prime factorization of 18: 2 × 3²
- Prime factorization of 24: 2³ × 3
- LCM(18, 24): Take the highest powers of 2 and 3. So, LCM = 2³ × 3² = 72.
- HCF(18, 24): Take the lowest powers of 2 and 3. So, HCF = 2 × 3 = 6.
Division Method
The Division Method is a simpler technique for finding the LCM and HCF, especially for larger numbers.
Steps: Add the digits of the number together, and check if the result is divisible by 3.
Steps to Find LCM:
- Divide the given numbers by their common prime factors.
- Repeat the process until no further division is possible.
- Multiply the divisors to get the LCM.
Steps to Find HCF:
- Keep dividing the larger number by the smaller number.
- Continue the process until the remainder is zero.
- The divisor at this step is the HCF.
Example:
Let’s take two numbers, 48 and 60.
Finding HCF:
- 60 ÷ 48 = 1 (remainder 12)
- 48 ÷ 12 = 4 (remainder 0)
So, HCF(48, 60) = 12.
Finding LCM:
- List the prime factors:
- Prime factorization of 48: 2⁴ × 3
- Prime factorization of 60: 2² × 3 × 5
- LCM = 2⁴ × 3 × 5 = 240.
Thus, LCM(48, 60) = 240.
Shortcut Method for Two Numbers
You can use a formula to find the LCM and HCF of two numbers using their product:
LCM(a,b) =[ a×b / HCF(a,b) ]
This formula comes in handy when you know either the LCM or HCF and need to find the other.
Example:
Let’s take the numbers 48 and 60 again.
- HCF(48, 60) = 12
- Using the formula:
LCM (48,60) = (48×60) / 12 = 240
Thus, LCM = 240.
Section 2
Applications of LCM and HCF
LCM and HCF aren’t just theoretical concepts; they are widely used in various fields and everyday problems. Let’s explore some of the most common applications:
Real-Life Applications of LCM
- Synchronizing Events: LCM helps in finding the time after which two or more events will happen simultaneously. For example, if two traffic lights blink every 40 seconds and 60 seconds, respectively, LCM will tell you when both lights will blink together.
- Packing Problems: If you need to pack items into boxes of different sizes and want to find the smallest number of items you can evenly distribute, LCM is the tool to use.
Example:
Suppose a bus arrives every 15 minutes and a train arrives every 20 minutes at the station. When will they both arrive together again?
- LCM (15, 20) = 60 minutes.
So, they will both arrive together after 60 minutes.
Real-Life Applications of HCF
- Dividing Items into Groups: HCF helps in finding how you can divide a number of items into the largest possible groups. For instance, if you have different-sized containers and you want to find the largest portion that can fit into all, HCF is used.
- Simplifying Fractions: HCF is used to simplify fractions by dividing the numerator and denominator by their HCF.
Example:
You have two pieces of ribbon, one 48 cm and the other 60 cm long, and you want to cut them into the longest possible equal-length pieces without any remainder. The length of each piece will be the HCF of 48 and 60.
- HCF(48, 60) = 12 cm.
Thus, you can cut the ribbons into 12 cm pieces.
MCQ Test
Number Systems Quiz
Here are 30 basic-level General Aptitude questions, each with multiple-choice options and detailed answers
Section 3
Practice Problems
Let’s go through a few problems to reinforce your understanding.
Find the LCM and HCF of 28 and 42.
- Prime factorization of 28: 2² × 7
- Prime factorization of 42: 2 × 3 × 7
- LCM(28, 42): Highest powers of 2, 3, and 7. So, LCM = 2² × 3 × 7 = 84.
- HCF(28, 42): Lowest powers of 2 and 7. So, HCF = 2 × 7 = 14.
Three numbers are 12, 15, and 18. Find their LCM and HCF.
- Prime factorization of 12: 2² × 3
- Prime factorization of 15: 3 × 5
- Prime factorization of 18: 2 × 3²
- LCM(12, 15, 18): Take the highest powers of 2, 3, and 5. So, LCM = 2² × 3² × 5 = 180.
- HCF(12, 15, 18): Common prime factor is 3. So, HCF = 3.
Section 4
Conclusion
Understanding LCM and HCF is essential for cracking various aptitude exams and solving real-life problems. While the concepts may seem abstract initially, with consistent practice and a structured approach, they become much simpler. By using methods such as prime factorization, the division method, and shortcuts like the LCM-HCF product formula, you can tackle even the toughest problems with ease.
Leave a Reply