Prime Factorization of HCF and LCM

Introduction to Prime Factorization of HCF and LCM 

The most important prime factor for HCF and LCM involves expressing the number of products of their prime factors. To find HCF, take main prime factors with the lowest powers; For LCM, take all the prime factors with the highest powers. For example, 60 = 2^2 × 3 × 5 & 48 = 2⁴ × 3. Therefore, HCF = 2^2 × 3 = 12 and LCM = 2⁴ × 3 × 5 = 240. This method simplifies calculations & helps to understand the number conditions effectively.

 

Table of Contents

• What is Prime Factorization?

• Definition of Prime Factorization

• Why Prime Factorization is Useful

• Prime Number Breakdown Examples

• Understanding HCF and LCM

  • What is the Highest Common Factor (HCF)?

  • What is the Least Common Multiple (LCM)?

• Comparison Table of HCF and LCM

• How to Find HCF Using Prime Factorization Method

• How to Find LCM Using Prime Factorization Method

• Difference Between HCF and LCM

• Solved Examples on Prime Factorization of HCF and LCM

• Practice Questions on HCF and LCM Using Prime Factors

• Conclusion
  
  
• Related Links

• FAQs on Prime Factorization of HCF and LCM
  
  

What is Prime Factorization?

The most important prime factorization is the process of expressing a number as a product of prime numbers. This plays an important role in understanding the properties, especially when two or more numbers calculate HCF (highest common factor) and LCM (the lowest common multiple). This method is useful for simplifying fractions, solving word problems and improving the sense of number.

Definition of prime factorization 

Prime factorization refers to a total number to break into a set of prime numbers, when multiplied together, provides the original number. A prime number has more than 1 number with only two factors - 1 and itself.

Example:

The prime factorization of 36 is: 36 = 2 × 2 × 3 × 3 = 2² × 3²

Why Prime Factorization is Useful

It will be easier to use prime factorization:

• HCF (Highest Common Factor) and LCM (lowest Common Factor) detects quickly
• Simple fractions
• Understand the structure of numbers
• Solve problems related to divisibility and multiples

Application:

• HCF is calculated by using common prime factors with the smallest powers.
• LCM is calculated using all the prime factors with the highest powers.

Prime Number Breakdown Examples

Here is the table showing the prime factorization of a few common numbers:

 

 

Understanding HCF and LCM

To effectively work with numbers in mathematics, it is important to understand the terms HCF (Highest Common Factor) and LCM (Least Common Multiple). These are necessary to compare the amount, solve the problems with the word and simplify mathematical manifestations. Learning to find HCF & LCM using prime factorization ensures accuracy and efficiency.

What is the Highest Common Factor (HCF)?

The Highest Common Factor (HCF) is the largest number sharing two or more given numbers without leaving the remainder. It is also known as the Greatest Common Divisor(GCD). HCF is found by identifying the common prime factors for numbers and taking the lowest powers to these factors.

Key points:

• HCF focuses on common divisors.
• Useful in simplifying fractions and ratios
• Helps dividing objects into equal parts
  
  

Example using Prime Factorization:

Let’s find the HCF of 60 and 48:

• Prime factorization of 60 = 2² × 3 × 5
• Prime factorization of 48 = 2⁴ × 3
• Common prime factors = 2² × 3
• HCF = 2² × 3 = 12

What is Least Common Multiple (LCM)?

The Least common multiple (LCM) Of the two or more numbers is the smallest number that is equally divided by all given numbers. To find LCM using prime factorization, we include all main prime factors using the highest powers from each number.

Key points:

• LCM focuses on multiples
• Normally time interval or plan is useful for finding
• It is important to add or subtracting as opposed to fractions

Example using Prime Factorization:

Find the LCM of 60 and 48:

• Prime factorization of 60 = 2² × 3 × 5
• Prime factorization of 48 = 2⁴ × 3
• All prime factors (take highest powers) = 2⁴ × 3 × 5
• LCM = 240
  
  

Comparison Table of HCF and LCM

 

How to Find HCF Using Prime Factorization Method

Detection of the highest common factor (HCF) using the Prime Factorization method is a simple and reliable technique that includes dividing each number into the most important prime  factors and then identifying the common ones. This method ensures accurate results, especially when working with large or complex numbers.

Steps to find HCF by prime factorization

Follow these simple steps to find HCF using Prime Factorization:

→ Step-by-step method:

1. Write the prime  factors for each number.
2. Identify common prime factors between the numbers.
3. Choose the lowest power for common prime factors.
4. Multiply the usual factors to get HCF.

Example:

Find the HCF of 36 and 48.

• 36 = 2² × 3²
• 48 = 2⁴ × 3
• Common prime factors = 2 and 3
• Lowest powers = 2² and 3
• HCF = 2² × 3 = 4 × 3 = 12

 

How to Find LCM Using Prime Factorization Method

The Least Common Multiple (LCM) helps us find the smallest multiple that is divided by two or more numbers. Using the prime factor to find LCM is especially useful in algebra, fractions and time -based problems.

Steps to find LCM at Prime Factorization

To find LCM by prime factorization method, use this structured process:

Step by step -method:

1. Write the prime factorization  for each number.
2. List all prime numbers that appear in any factorization.
3. Choose the highest power for each prime number.
4. Multiply them together to get LCM.

Example:

Find the LCM of 36 and 48.

• 36 = 2² × 3²
• 48 = 2⁴ × 3
• All prime factors involved = 2 and 3
• Highest powers = 2⁴ and 3²
• LCM = 2⁴ × 3² = 16 × 9 = 144
  
  

Difference Between HCF and LCM

 

 

Solved Examples on Prime Factorization of HCF and LCM

Practicing examples helps students clearly understand the prime factorization method for HCF and LCM. The examples below show how to use a step-by-step approach to both the highest common factor (HCF) and at least common multiple (LCM) when using prime factors. Each example shows how the number of breakdowns in the prime components can find HCF and LCM easily and accurately.

Example 1: Find HCF Using Prime Factors

Problem: Find the HCF of 72 and 108 using prime factorization.

Solution: Step-by-step prime factorization:

• 72 = 2³ × 3²
• 108 = 2² × 3³

Common prime factors: 2 and 3

Lowest powers: 2² and 3²

HCF = 2² × 3² = 4 × 9 = 36

Example 2: Find LCM Using Prime Factorization

Problem: Find the LCM of 18 and 30 using the prime factor method.

Solution: Prime factorization:

• 18 = 2 × 3²
• 30 = 2 × 3 × 5

All prime factors involved: 2, 3, and 5

Highest powers: 2¹, 3², and 5¹

LCM = 2 × 3² × 5 = 2 × 9 × 5 = 90

Example 3: Find HCF and LCM of Two Numbers

Problem: Find both HCF and LCM of 40 and 64 using prime factorization.

Solution:

Step 1: Prime factorization

• 40 = 2³ × 5
• 64 = 2⁶

HCF:

• Common prime factor = 2
• Lowest power = 2³
• HCF = 8

LCM:

• Prime factors: 2 (take 2⁶) and 5
• LCM = 2⁶ × 5 = 64 × 5 = 320
  
  

Practice Questions on HCF and LCM Using Prime Factors

1. Find the HCF of 36 & 60 using prime factorization.
2. Find the LCM of 24 and 36 using prime factors.
3. Find the LCM & HCF of 45 and 75 using the prime factors method.
4. Three numbers are 30, 45 & 60. Find their HCF and LCM.
5. Using prime factorization, simplify the ratio of 56 and 84.
   
   

Conclusion

Prime factorization is a powerful and reliable way to find HCF and LCM for any set of Prime Factorification numbers. By breaking numbers in their basic building blocks (prime factors), their structure becomes easy to understand and quickly easy to identify normal factors (for HCF) or combined factors (for LCM). This method not only simplifies calculations, but also strengthens your understanding of the numbers' relation.Mastering  it will be much more controlled and nice to handle problems related to HCF, LCM and division.

 

Related Links

• LCM Factor – Learn how to find the least common multiple of numbers, its significance, and examples for everyday math problems.
• HCF and LCM – Understand the concepts of HCF and LCM, their differences, and how to apply them to solve math questions effectively.
• HCF Factor – Discover the role of HCF in math, with simple explanations and examples to find the highest common factor with ease.
  
  

FAQs on Prime Factorization of HCF and LCM

1. What is the formula for HCF and LCM?
   The relationship between the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two numbers is that their product is equal to the product of the two numbers themselves. In other words, for any two numbers, say 'a' and 'b', LCM(a, b) * HCF(a, b) = a * b. This formula allows you to find one if you know the other and the two numbers.

2. How to do LCM and HCF by prime factorisation method?
   To find the Least Common Multiple (LCM) and Highest Common Factor (HCF) of numbers using prime factorization, first, express each number as a product of its prime factors. Then, for the HCF, identify the common prime factors and multiply the smallest power of each. For the LCM, multiply all prime factors raised to their highest powers found in any of the numbers

3. What is the HCF & LCM of 108-120 and 252 using the prime factorization method?

• CF ( 108,120,252 )= product of common terms with lowest power = ( 22 × 3 ) = ( 4 × 3 ) = 12 . 
• CM ( 108,120,252 ) = product of prime factors with highest power = ( 23 × 33 × 5 × 7 ) = 7560. 
• HCF=12 and LCM=7560. find the largest number which divides 245 and 1037, leaving the remainder 5 in each case.

4. How do you factor HCF and LCM?
   To find the HCF, find any prime factors that are in common between the products. Each product contains two 2s and one 3, so use these for the HCF. Cross any numbers used so far off from the products. To find the LCM, multiply the HCF by all the numbers in the products that have not yet been used.

5. What is the HCF and LCM of 72 126 and 168 using prime factorization?
   HCF=6,LCM=504. Step by step video, text & image solution for Using prime factorization method find the HCF and LCM of 72 , 126 and 168.

 

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