HCF and LCM

Mathematics becomes easier when students understand basic number concepts clearly. One of the most important topics in arithmetic is HCF and LCM. These concepts are widely used in school mathematics, competitive exams, and even in real-life situations such as arranging schedules, grouping items, and solving numerical problems.

The HCF (Highest Common Factor) helps us find the greatest number that divides two or more numbers exactly, while the LCM (Least Common Multiple) helps us find the smallest number that is divisible by two or more numbers.

Understanding HCF and LCM improves problem-solving skills and builds a strong foundation for advanced mathematical concepts. In this article, we will learn the meaning, formulas, methods, examples, and applications of HCF and LCM in a simple and student-friendly way.

1.0What is HCF?

The full form of HCF is Highest Common Factor. It is the largest number that divides two or more numbers exactly without leaving any remainder. In simple words, HCF is the greatest common factor shared by the given numbers.

Example of HCF

Find the HCF of 12 and 18.

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

Common factors = 1, 2, 3, 6

The greatest common factor is 6.

Therefore,
HCF of 12 and 18 = 6

HCF is also known as:

• Greatest Common Divisor (GCD)
• Greatest Common Factor (GCF)

2.0What is LCM?

The full form of LCM is Least Common Multiple. It is the smallest number that is exactly divisible by two or more numbers.

In simple terms, LCM is the first common multiple shared by the given numbers.

Example of LCM

Find the LCM of 4 and 6.

Multiples of 4 = 4, 8, 12, 16, 20...
Multiples of 6 = 6, 12, 18, 24...

The first common multiple is 12.

Therefore,
LCM of 4 and 6 = 12

LCM is very useful in solving timetable problems, fractions, and synchronization questions.

3.0Difference Between HCF and LCM

Understanding the difference between HCF and LCM helps students solve mathematical problems more accurately.

4.0Methods of Finding HCF and LCM

There are different methods of finding HCF and LCM. Let us understand them one by one.

1. Prime Factorization Method

The prime factorization method is one of the easiest ways to find HCF and LCM.

Find the HCF of 24 and 36.

Prime factors of 24 = 2 × 2 × 2 × 3
Prime factors of 36 = 2 × 2 × 3 × 3

Common prime factors = 2 × 2 × 3 = 12

Therefore, HCF = 12

Finding LCM Using Prime Factorization

Find the LCM of 12 and 18.

Prime factors of 12 = 2 × 2 × 3
Prime factors of 18 = 2 × 3 × 3

Take the highest powers of all prime factors: LCM = 2 × 2 × 3 × 3 = 36

Therefore, LCM = 36

2. Division Method

The division method is commonly used to find HCF quickly.

Example: Find the HCF of 48 and 18.

48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6 = 2 remainder 0

The last divisor is 6. Therefore, HCF = 6

5.03. Listing/Common Multiples Method

This method is useful for smaller numbers.

Example: Find the LCM of 5 and 7.

Multiples of 5 = 5, 10, 15, 20, 25, 30, 35...
Multiples of 7 = 7, 14, 21, 28, 35...

The first common multiple is 35. Therefore, LCM = 35

6.0Formula Related to HCF and LCM

For any two numbers:

HCF × LCM = Product of numbers

This formula is very important in mathematics and competitive exams.

Numerical Example

Find the LCM of 8 and 12 if HCF = 4.

Using the formula: HCF × LCM = Product of numbers

4 × LCM = 8 × 12
4 × LCM = 96
LCM = 96 ÷ 4
LCM = 24

Therefore, LCM = 24

7.0Real-Life Applications of HCF and LCM

The concepts of HCF and LCM are not limited to textbooks. They are used in many real-life situations.

1. Grouping and Distribution: HCF helps in dividing items into equal groups without leftovers.

Example: If 24 chocolates and 36 candies are to be packed equally, HCF helps determine the maximum number of packets.

2. Timetable Synchronization: LCM helps find when repeating events happen together.

Example: Two school bells ring every 15 minutes and 20 minutes. Their LCM gives the time when both bells ring together again.

3. Fractions: LCM is used while adding or subtracting fractions with different denominators.

8.0The Key Properties of HCF and LCM

1. Value Boundaries Property

• HCF Boundary: The HCF of any given set of numbers is never greater than the smallest number in that set.
• LCM Boundary: The LCM of any given set of numbers is never less than the largest number in that set.

2. The Factor-Multiple Relationship Property

• Divisibility: The HCF of a group of numbers is always a perfect factor of their LCM. This means LCM / HCF will always yield a whole integer with zero remainder.

3. The Product Property (Strictly for Pairs)

• Two-Number Rule: For exactly two natural numbers (let's call them A and B), the product of their HCF and LCM is always equal to the product of the two numbers themselves:
  HCF x LCM = A x B
• Limitation: This property does not work for a group of three or more numbers.

4. Co-Prime Numbers Property

Co-prime numbers (like 8 and 15, or any two consecutive integers like 12 and 13) share no common factors other than 1.

• HCF of Co-Primes: Always equals 1.
• LCM of Co-Primes: Always equals the product of the numbers themselves (A x B).

5. Multiples and Factors Property

• If one number is a perfect factor of another number (for example, 5 and 20, where 5 is a factor of 20):
• • Their HCF is the smaller number (5).
  • Their LCM is the larger number (20).

6. Fractions Property

When finding the HCF and LCM of a group of fractions, the calculations are split between the numerators (top numbers) and denominators (bottom numbers):

• HCF of Fractions = (HCF of Numerators) / (LCM of Denominators)
• LCM of Fractions = (LCM of Numerators) / (HCF of Denominators)

9.0Solved Examples

Question: Find the HCF and LCM of 24 and 60 using the prime factorization method.

Solution: Step 1: Find the Prime Factorization of Each Number

First, break down both numbers into their foundational prime factors expressed in exponent form:

• 24 = 2 x 2 x 2 x 3 = 2³ x 3¹ 
• 60 = 2 x 2 x 3 x 5 = 2² x 3¹ x 5¹ 

Step 2: Calculate the Highest Common Factor (HCF)

Rule for HCF: The HCF is the product of only the common prime factors, taking the least (lowest) power of each common factor.

• The common prime factors appearing in both lists are 2 and 3. (Note: 5 is left out because it does not appear in the factorization of 24).
• The lowest power of the common factor 2 is 2².
• The lowest power of the common factor 3 is 3¹.

Multiply these lowest powers together to isolate the HCF:

HCF = 2² x 3¹ = 4 x 3 = 12 

Step 3: Calculate the Lowest Common Multiple (LCM)

Rule for LCM: The LCM is the product of all prime factors involved across both numbers, taking the highest power of each factor.

• The prime factors involved across both numbers are 2, 3, and 5.
• The highest power of 2 present is 2³.
• The highest power of 3 present is 3¹.
• The highest power of 5 present is 5¹.

Multiply these highest powers together to find the LCM:

LCM = 2³ x 3¹ x 5¹ = 8 x 3 x 5 = 120

Example 2: Calculating HCF and LCM for Fractions

Question: Find the exact HCF and LCM for the fraction set: 4/9 and 6/21.

Solution:

• Step 1: Group the numerators and denominators.Numerators = 4 and 6Denominators = 9 and 21
• Step 2: Find the individual HCF and LCM components for these groups.Factors of 4 = 2 x 2, Factors of 6 = 2 x 3 -> HCF of Numerators = 2Multiples of 4 and 6 -> LCM of Numerators = 12Factors of 9 = 3 x 3Factors of 21 = 3 x 7 -> HCF of Denominators = 3Multiples of 9 and 21 -> LCM of Denominators = 63
• Step 3: Apply the fraction formulas.HCF of Fractions = (HCF of Numerators) / (LCM of Denominators) = 2/63LCM of Fractions = (LCM of Numerators) / (HCF of Denominators) = 12/3 = 4

Answer: The HCF of the fractions is 2/63 and their LCM is 4.