Understanding HCF and LCM: A Simple Guide

Factors and Multiples

Factors are numbers that divide another number exactly, without leaving a remainder. Conversely, if a number can be divided exactly by another number, that second number is called a multiple of the first number.

Example:

• Consider the number 12. Its factors are 1, 2, 3, 4, 6, and 12.
• The number 4 has multiples like 4, 8, 12, 16, and so on.

If you divide 12 by 4, you get 3 with no remainder, so 4 is a factor of 12, and 12 is a multiple of 4.

When you start learning math, you come across two important concepts called HCF and LCM. These might sound complicated, but they are quite simple once you get the hang of them. Let’s break them down in a way that’s easy to understand!

What is HCF?

HCF stands for Highest Common Factor. Sometimes it’s also called GCD, which means Greatest Common Divisor. But don’t worry about the fancy names—what matters is what it does!

HCF is the largest number that can evenly divide two or more numbers. Think of it as the biggest piece that can fit into all the numbers you’re looking at.

Example: Imagine you have 12 apples and 15 oranges. You want to make fruit baskets so that each basket has the same number of apples and oranges, and you don’t want any fruit left over. What’s the largest number of fruits each basket can have?

To find this, you look at the factors of each number:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 15: 1, 3, 5, 15

The common factors are 1 and 3. The biggest common factor is 3. So, the HCF of 12 and 15 is 3. This means you can make baskets with 3 apples and 3 oranges each, and you will use up all the fruits!

What is LCM?

LCM stands for Least Common Multiple. It’s the smallest number that is a multiple of two or more numbers. In simple terms, it’s the smallest number that both numbers can divide into without any remainder.

Example: Let’s use the same numbers—12 and 15. You want to find the smallest number where you can group them evenly. What’s the smallest number that both 12 and 15 can divide into?

To find the LCM, list the multiples of each number:

• Multiples of 12: 12, 24, 36, 48, 60, 72, …
• Multiples of 15: 15, 30, 45, 60, 75, …

The smallest number that appears in both lists is 60. So, the LCM of 12 and 15 is 60. This means if you have to arrange items in groups of 60, both 12 and 15 can fit into this arrangement evenly.

How to Find HCF and LCM

Finding HCF:

1. List the Factors: Write down all the factors of each number.
2. Find the Common Factors: Look at which factors are common.
3. Choose the Largest: Pick the biggest common factor. That’s your HCF!

Finding LCM:

1. List the Multiples: Write down multiples of each number.
2. Find the Smallest Common Multiple: Look at which multiples are common and pick the smallest one. That’s your LCM!

Methods to Find HCF

1. Prime Factorization Method

   • Step 1: Factorize each number into its prime factors.
   • Step 2: Identify the common prime factors.
   • Step 3: Multiply the common prime factors to get the HCF.

          Example: What is the HCF of 18 and 24.

   • 18 = 2 × 3²
   • 24 = 2³ × 3
   • Common prime factors: 2¹ × 3¹
   • HCF = 2¹ × 3¹ = 6

2. Division Method (Euclidean Algorithm)

   • Step 1: Divide the larger number by the smaller number and find the remainder.
   • Step 2: Replace the larger number with the smaller number and the smaller number with the remainder.
   • Step 3: Repeat the process until the remainder is 0. The divisor at this stage is the HCF.

            Example: Find the HCF of 56 and 98.

   • 98 ÷ 56 = 1 remainder 42
   • 56 ÷ 42 = 1 remainder 14
   • 42 ÷ 14 = 3 remainder 0
   • HCF = 14

3. Listing Factors

   • Step 1: List all factors of each number.
   • Step 2: Identify the common factors.
   • Step 3: Choose the largest common factor.

           Example: Find the HCF of 15 and 25.

   • Factors of 15: 1, 3, 5, 15
   • Factors of 25: 1, 5, 2
   • Common factors: 1, 5
   • HCF = 5

Methods to Find LCM

1. Prime Factorization Method

   • Step 1: Factorize each number into its prime factors.
   • Step 2: For each prime factor, take the highest power that appears in the factorizations.
   • Step 3: Multiply these highest powers to get the LCM.

            Example: Find the LCM of 15 and 20.

   • 15 = 3 × 5
   • 20 = 2² × 5
   • Highest powers: 2² × 3¹ × 5¹
   • LCM = 2² × 3¹ × 5¹ = 60

2. Listing Multiples

   • Step 1: List some multiples of each number.
   • Step 2: Identify the smallest common multiple.

         Example: Find the LCM of 4 and 6.

   • Multiples of 4: 4, 8, 12, 16, 20
   • Multiples of 6: 6, 12, 18, 24
   • Smallest common multiple: 12
   • LCM = 12

3. Using HCF (Greatest Common Divisor)

   • Step 1: Find the HCF of the numbers.
   • Step 2: Use the formula:
     $$\text{LCM}(a, b) = \frac{|a \times b|}{\text{HCF}(a, b)}$$ 

     ​

         Example: Find the LCM of 8 and 12 using HCF.

   • HCF of 8 and 12 = 4
   • LCM =
     $\frac{8 \times 12}{4} = 24$ 

Quick Tips on HCF and LCM

• HCF is about breaking things down into their biggest common pieces.
• LCM is about finding the smallest number that can be evenly divided by the numbers you have.

Product of Two Numbers

The product of two numbers can be found using their HCF and LCM:

$$\text{Product of Two Numbers} = \text{HCF} \times \text{LCM}$$

Co-primes

Two numbers are co-prime if their HCF is 1. This means they have no common factors other than 1.

Example: The numbers 8 and 15 are co-prime because their HCF is 1.

HCF and LCM of Fractions

HCF of Fractions:

$$\text{HCF} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$$

​

LCM of Fractions:

$$\text{LCM} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$$

​

HCF and LCM of Decimal Fractions

1. Convert decimal fractions to fractions with the same number of decimal places.
2. Find the HCF or LCM of these fractions.
3. Convert the result back to decimal form, adjusting the number of decimal places.

Comparison of Fractions

1. Find the LCM of the denominators of the given fractions.
2. Convert each fraction to an equivalent fraction with the LCM as the denominator.
3. Compare the numerators of these equivalent fractions. The fraction with the greatest numerator is the largest.

Example: To compare

$\frac{2}{5}$

and

$\frac{3}{7}$

:

• LCM of 5 and 7 is 35.
• Convert fractions:
  $\frac{2}{5} = \frac{14}{35}$ 

  ​ and $\frac{3}{7} = \frac{15}{35}$ 

  ​.

• Compare numerators: 14 and 15. Thus,
  $\frac{3}{7}$ is greater.

Practice Makes Perfect!

The best way to get good at finding HCF and LCM is to practice with different sets of numbers. Start with small numbers and gradually try more challenging ones. You’ll get better and faster with time!

Sample Questions on HCF and LCM

1. Find the HCF of 36 and 48.
   • Method: Use Prime Factorization.
   • Solution: 36 = 2² × 3², 48 = 2⁴ × 3¹
     • Common factors: 2² × 3¹ = 12
   • Answer: HCF = 12
2. Find the LCM of 9 and 15.
   • Method: Use Prime Factorization.
   • Solution: 9 = 3², 15 = 3¹ × 5¹
     • Highest powers: 3² × 5¹ = 45
   • Answer: LCM = 45
3. Find the HCF of 54 and 72 using the Division Method.
   • Solution:
     • 72 ÷ 54 = 1 remainder 18
     • 54 ÷ 18 = 3 remainder 0
   • Answer: HCF = 18
4. Find the LCM of 7 and 14 using the HCF method.
   • Method:
     • HCF of 7 and 14 = 7
     • LCM =
       $\frac{7\times 14}{7}=\mathrm{14<span\; style="background-color:\; var(--wp--preset--color--base);\; font-family:\; var(--wp--preset--font-family--roboto);\; font-size:\; var(--wp--preset--font-size--small);"> </span>}$
   • Answer: LCM = 14

By practicing these methods and questions, you’ll get more comfortable with finding HCF and LCM.

Frequently Asked Questions (FAQ)

1. What are factors and multiples?

Answer: Factors are numbers that divide another number exactly without leaving a remainder. For example, factors of 12 are 1, 2, 3, 4, 6, and 12. Multiples are numbers that can be divided exactly by a given number. For example, multiples of 4 include 4, 8, 12, 16, and so on.

2. What is the Highest Common Factor (HCF)?

Answer: The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD) or Greatest Common Measure (GCM), is the largest number that divides two or more numbers exactly.

Example: For the numbers 18 and 24, the HCF is 6 because 6 is the largest number that divides both 18 and 24 without leaving a remainder.

3. How do you find the HCF using the Factorization Method?

Answer:

1. Express each number as a product of prime factors.
2. Identify the common prime factors among the numbers.
3. Multiply the lowest powers of these common factors to get the HCF.

Example: For 18 (2 × 3²) and 24 (2³ × 3¹):

• Common prime factors: 2¹ × 3¹
• HCF = 2 × 3 = 6

4. What is the Division Method for finding HCF?

Answer: The Division Method (Euclidean Algorithm) involves:

1. Dividing the larger number by the smaller number and noting the remainder.
2. Replacing the larger number with the smaller number and the smaller number with the remainder.
3. Repeating until the remainder is 0. The last non-zero remainder is the HCF.

Example: For 56 and 98:

• 98 ÷ 56 = 1 remainder 42
• 56 ÷ 42 = 1 remainder 14
• 42 ÷ 14 = 3 remainder 0
• HCF = 14

5. What is the Least Common Multiple (LCM)?

Answer: The Least Common Multiple (LCM) is the smallest number that is exactly divisible by each of the given numbers.

Example: For 9 and 15:

• Prime Factorization:
  • 9 = 3²
  • 15 = 3¹ × 5¹
• Highest Powers: 3² and 5¹
• LCM = 3² × 5 = 45

6. How do you find the LCM using the Factorization Method?

Answer:

1. Express each number as a product of prime factors.
2. Take the highest power of each prime factor that appears in the factorizations.
3. Multiply these highest powers to get the LCM.

7. What is the Division Method for finding LCM?

Answer:

1. Arrange the numbers in a row.
2. Divide by a number that divides at least two of the numbers, then continue with the results.
3. The product of all the divisors and undivided numbers gives the LCM.

Example: For 8 and 12:

• Divide by 2: 8 → 4, 12 → 6
• Divide by 2 again: 4 → 2, 6 → 3
• Divide by 2 once more: 2 → 1, 3 → 3
• Product of divisors and final numbers = 2 × 2 × 2 × 3 = 24
• LCM = 24

8. How do you find the product of two numbers using HCF and LCM?

Answer: The product of two numbers is equal to the product of their HCF and LCM.

$$\text{Product of Two Numbers}=\text{HCF}\times \text{LCM}$$

9. What are co-prime numbers?

Answer: Two numbers are co-prime if their HCF is 1. This means they have no common factors other than 1.

Example: The numbers 8 and 15 are co-prime because their HCF is 1.

10. How do you find the HCF and LCM of fractions?

Answer:

• HCF of Fractions:

  $$\text{HCF}=\frac{\text{HCF of Numerators}}{\text{LCM of Denominators<span style="font-family: Georgia, 'Times New Roman', 'Bitstream Charter', Times, serif;"></span>}}$$

• LCM of Fractions:

  $$\text{LCM}=\frac{\text{LCM of Numerators}}{\text{HCF of Denominators<span style="font-family: Georgia, 'Times New Roman', 'Bitstream Charter', Times, serif;"></span>}}$$

11. How do you find the HCF and LCM of decimal fractions?

Answer:

1. Convert decimal fractions to fractions with the same number of decimal places.
2. Find the HCF or LCM of these fractions.
3. Adjust the result back to decimal form, matching the number of decimal places.

12. How do you compare fractions?

Answer:

1. Find the LCM of the denominators of the given fractions.
2. Convert each fraction to an equivalent fraction with the LCM as the denominator.
3. Compare the numerators of these equivalent fractions. The fraction with the greatest numerator is the largest.

Example: To compare $\frac{2}{5}$ and $\frac{3}{7}$:

• LCM of 5 and 7 is 35.
• Convert fractions: $\frac{2}{5} = \frac{14}{35}$ and $\frac{3}{7} = \frac{15}{35}$.
• Compare numerators: 14 and 15. Thus, $\frac{3}{7}$ is greater.