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HCF and LCM

Introduction to HCF and LCM

In mathematics, HCF and LCM are foundational concepts that are frequently used in arithmetic operations, algebraic expressions, and real-life problem-solving. HCF stands for Highest Common Factor, and LCM stands for Least Common Multiple. Understanding how to calculate HCF and LCM is essential for solving problems involving time intervals, work-sharing, and simplification of fractions.

This will help you learn the LCM and HCF formulas, step-by-step methods, and real-world LCM and HCF examples to strengthen your concepts.

 

Table of Contents

 

HCF Definition

HCF (Highest Common Factor) is the greatest number that divides two or more given numbers exactly without leaving any remainder.

Example:

Find the HCF of 24 and 36.

  • Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24

  • Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36

  • Common factors = 1, 2, 3, 4, 6, 12

  • HCF = 12

 

LCM Definition

LCM (Least Common Multiple) is the smallest number that is a common multiple of two or more numbers. It is especially helpful when dealing with time cycles, calendar calculations, and the addition or subtraction of unlike fractions.

Example:

Find the LCM of 5 and 7.

  • Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40

  • Multiples of 7 = 7, 14, 21, 28, 35, 42

  • LCM = 35

 

LCM of Two Numbers

To find the LCM of two numbers, determine the smallest number that is divisible by both. This is useful when synchronizing events or combining repetitive cycles.

Example:

LCM of 8 and 12 is 24.

 

HCF and LCM Formula

Two key ideas in number theory that are helpful in our understanding of the relationship between numbers in terms of divisibility and multiples are the Highest Common Factor (HCF) and the Least Common Multiple (LCM). The largest number that precisely divides all of the given numbers without leaving a remainder is the HCF of two or more numbers. On the other hand, the smallest number that is a multiple of all the given numbers is the LCM of two or more numbers.

The following formula provides one of the most significant connections between the HCF and LCM of two numbers:

HCF × LCM = Product of the Numbers

HCF and LCM Formula, If a and b are two numbers:

HCF(a, b) × LCM(a, b) = a × b

Example:

Find the HCF and LCM of 10 and 15

  • HCF = 5

  • LCM = 30

  • Product = 10 × 15 = 150

  • HCF × LCM = 5 × 30 = 150

This formula is helpful for cross-checking your answers.

 

Methods to Calculate HCF and LCM

HCF by Prime Factorization Method

  • Write the prime factors of each number

  • Identify the common prime factors

  • Multiply them to get the HCF

Example: HCF of 30 and 45

  • 30 = 2 × 3 × 5

  • 45 = 3 × 3 × 5

  • Common prime factors = 3 × 5 = 15

HCF by Division Method

  • Divide the larger number by the smaller

  • Divide the divisor by the remainder

  • Continue until the remainder is zero

  • The last divisor is the HCF

Example: HCF of 48 and 18

  • 48 ÷ 18 = 2 remainder 12

  • 18 ÷ 12 = 1 remainder 6

  • 12 ÷ 6 = 2 remainder 0

  • HCF = 6

LCM by Prime Factorization Method

  • Write the prime factorization of all numbers

  • Take all prime factors with the highest power

  • Multiply them together

Example: LCM of 8 and 12

  • 8 = 2³

  • 12 = 2² × 3

  • LCM = 2³ × 3 = 24

LCM by Division Method

  • Write all numbers in a row

  • Divide by the smallest prime number

  • Continue dividing until all values become 1

  • Multiply all the divisors to get the LCM

Example: LCM of 6 and 8

  • 2 | 6, 8

  • 2 | 3, 4

  • 2 | 3, 2

  • 3 | 3, 1

  • Result = 2 × 2 × 2 × 3 = 24

 

HCF and LCM Examples


Numbers

HCF

LCM

10, 15

5

30

12, 18

6

36

4, 6

2

12

7, 3

1

21

16, 20

4

80

These HCF and LCM examples help reinforce your understanding of both methods.

LCM and HCF for Fractions

Understanding LCM and HCF for fractions is important in solving fraction-based problems.

HCF of Fractions:

HCF = HCF of Numerators / LCM of Denominators

LCM of Fractions:

LCM = LCM of Numerators / HCF of Denominators

Example:

Find the HCF and LCM of 3/4 and 6/8

  • HCF (numerators) = HCF(3, 6) = 3

  • LCM (denominators) = LCM(4, 8) = 8

  • HCF = 3/8

  • LCM (numerators) = LCM(3, 6) = 6

  • HCF (denominators) = HCF(4, 8) = 4

  • LCM = 6/4 = 3/2

 

Common Misconceptions About HCF and LCM

Building a solid foundation in number theory requires an understanding of HCF and LCM. Here are some frequently misinterpreted concepts:

  • Misconception: HCF and LCM are Always Different Numbers

    • While usually different, HCF and LCM can sometimes be the same, particularly when the given numbers are equal.
      Example: For 6 and 6,

    • HCF = 6

    • LCM = 6

  • Misconception: LCM is Always Greater Than HCF

    • This is typically true, but not always. In some cases, the LCM and HCF relationship can be surprising, especially with 1 as one of the numbers.
      Example: For 1 and 10,

    • HCF = 1

    • LCM = 10

    • But if both numbers are 1, then both HCF and LCM are 1.

  • Misconception: The Product of HCF and LCM is the Product of the Numbers in All Cases

    • This property only applies to two numbers. It doesn't hold when more than two numbers are involved.
      Formula:
      For two numbers, a and b:
      HCF × LCM = a × b

  • Misconception: LCM and HCF Only Apply to Whole Numbers

    • Although HCF and LCM for fractions are most frequently applied to whole numbers, they are a legitimate and practical idea, particularly in advanced mathematics and real-world ratio-related problems.

 

Fun Facts About HCF and LCM

Make learning about HCF and LCM enjoyable with these interesting facts:

  • LCM and HCF Are Closely Connected
    For any two positive integers:
    HCF × LCM = Product of the Numbers
    This is one of the most significant LCM and HCF formulas, demonstrating their interdependence.

  • HCF of Co-Prime Numbers Is Always 1
    Two numbers are said to be co-prime if they share no factors other than 1. Their LCM is just the product of the two, and their HCF is always 1.

  • HCF and LCM in Real Life

    • HCF is useful when splitting things into smaller sections (like equally dividing a rope).

    • LCM helps in finding common periods or schedules (like syncing traffic lights or class timetables).

  • HCF and LCM for Fractions
    While it might seem counterintuitive, you can calculate LCM and HCF for fractions using formulas:
    • HCF of fractions = HCF of numerators / LCM of denominators
    • LCM of fractions = LCM of numerators / HCF of denominators

 

Conclusion

It is essential to grasp the ideas of HCF and LCM in order to solve algebraic, arithmetic, and real-world problems. Students can confidently tackle challenging problems by mastering the division and prime factorization methods for calculating HCF and LCM.

To develop a strong mathematical foundation, be sure to practice LCM and HCF examples, apply the LCM and HCF formulas, and use them to solve fraction, time, and work problems.

 

Related Links

LCM (Least Common Multiple)  - Dive into our complete guide on LCM for Class 4 students with simple explanations and examples.

Math Calculator - Use our all-in-one Math Calculator to solve problems quickly

 

FAQ’S

1. What is the LCM and HCF?

Two essential ideas in mathematics are HCF and LCM:

  • The largest number that precisely divides two or more numbers without leaving a remainder is known as the HCF (Highest Common Factor).

  • The smallest number that is a multiple of two or more numbers is called the LCM (Least Common Multiple).

Solving time and work problems, simplifying fractions, and other tasks all depend on knowing how to compute HCF and LCM.

 

2. What is the HCF of 24 and 36?

Utilise the prime factorization method to determine the HCF of 24 and 36:

  • 24's prime factors are 2 × 2 × 2 × 3.

  • 36 prime factors are 2 × 2 × 3 × 3.

  • Typical factors: 2 × 2 × 3 = 12.

12 is the HCF of 24 and 36.

3. How to find the HCF of 420 and 1782?

Use the division method, also known as the Euclidean algorithm, to determine the HCF of 420 and 1782:

Step 1: Divide the bigger number by the smaller one.

1782 ÷ 420 = 4, with 102 as the remainder

Step 2: Calculate the remainder by dividing the previous divisor (420):

420 ÷ 102 = 4 leftover Twelve

Step 3: Divide 102 by 12: 102 ÷ 12 = 8, leaving 6 as the remainder.

Step 4: Divide 12 by 6: 12 ÷ 6 = 2, with 0 as the remainder.

The HCF is the final divisor when the remainder equals zero.

Thus, the HCF of 1782 and 420 is 6.

 

4. What is the rule for HCF and LCM?

For HCF and LCM, the most crucial rule is:

HCF × LCM = The product of the two numbers 

This rule, which is frequently used to cross-verify results, is only valid for two numbers. If you already know the other value and the product of the numbers, it makes it easier to find one value quickly.


Keep practicing HCF and LCM problems to strengthen your skills. And continue exploring maths concepts with Orchids The International School!

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