Prime Factorization of HCF and LCM

Introduction to Prime Factorization of HCF and LCM 

Prime factorization means expressing a number as a product of its prime factors. For example, 24 can be written as 2 × 2 × 2 × 3. This method is very useful for finding HCF (Highest Common Factor) and LCM (Lowest Common Multiple) numbers.

HCF of two or more numbers is the largest number that divides them all, while LCM is the smallest number that is a multiple of all given numbers. Using prime factorization makes it easy to find both. To get HCF, we take the general prime factor. To get LCM, we take all prime factors, including the highest powers.

For example, for 18 and 24: The prime factors of 18 = 2 × 3 × 3, the prime factors of 24 = 2 × 2 × 2 × 3. Here, HCF = 2 × 3 = 6 and LCM = 2 × 2 × 2 × 3 × 3 = 72. In this way, Prime Factorization gives us a simple step-by-step method to calculate HCF and LCM.

 

Table of Contents

• Definition of Prime Factorization
• Highest Common Factor
• Least Common Multiple
• HCF by Prime Factorization Method
• HCF by Division Method
• LCM by Prime Factorization Method
• LCM by Division Method
• Solved Examples
• Practice Questions
• Conclusion
• FAQs on Prime Factorization of HCF and LCM

 

Definition of Prime Factorization

Prime factorization means breaking a number into smaller numbers that are prime. A prime number is a number that can only be divided by 1 and itself, like 2,3,5,7,11, etc, so the prime factorization is writing a number as a product of prime numbers. 

Example:

• The prime factorization of 12 is:

• 12 = 2 × 2 × 3. Here, 2 and 3 are prime numbers.

Prime Number Breakdown Examples

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

 

Highest Common Factor (HCF)

The Highest Common Factor (HCF) is the largest number that divides two or more given numbers without leaving a 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 of these factors.

Key points:

• HCF focuses on common divisors.

• Useful in simplifying fractions and ratios

• Helps divide 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

 

Least Common Multiple (LCM)

The Least common multiple (LCM) of two or more numbers is the smallest number that is evenly 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 a plan is useful for finding

• It is important to add or subtract 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

HCF by 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 its 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

HCF by Division Method

HCF (Highest Common Factor) of two or more numbers is the largest number that exactly divides them all (without leaving a reminder). In the division method, we divide the large number by small numbers and continue the process until the remaining is zero. The last division is HCF.

Example: Find the HCF of 42 and 30 

• Step 1: Divide the large numbers by the smallest number. Here, the larger number is 42 & the smaller is 30.

• 42  ÷ 30 = 1 

• Remainder is 12

• Step 2: Now, take the divisor 30 as the new dividend and the remainder 12 as the new divisor.

• 30 ÷ 12 = 2 

• Remainder is 6 

• Step 3: Again, take the divisor 12 as the new dividend and the remainder 6 as the new divisor.

• 12 ÷ 6 = 2

• Remainder is 0

• Step 4: When the remainder is 0, the last divisor is the HCF.

• So, the HCF of 42 & 30 is 6

 

LCM by 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 the 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

LCM by Division Method

The LCM ( Least Common Multiple) of two or more numbers is the smallest number that is a multiple of all the given numbers. In the division method, we divide the numbers by prime numbers (2, 3, 5, 7, ...) until we can’t divide further. Then we multiply all the prime numbers and the last row numbers to get LCM.

Example: Find the LCM of 18,24, and 30

Step 1: Write the numbers in one row: 18,24,30

Step 2: Start dividing by the smallest prime number (2):

• 18 ÷ 2 = 9

• 24 ÷ 2 = 12

• 30 ÷ 2 = 15

Now the row is 9,12,15

Step 3: Again, divide by 2 

• 9 - it's not divisible by 2 

• 12 ÷ 2 = 6

• 15 - it's not divisible by 2

Step 4: Again, divide by 2

• 9

• 6 ÷ 2 = 3

• 15

Now the row is 9,3,15

Step 5: Divide by 3

• 9 ÷ 3 = 3

• 3 ÷ 3 = 1

• 15 ÷ 3 = 5

Now the row is 3,1,5

Step 6: Again, divide by 3

• 3 ÷ 3 = 1

• 1

• 5

Now the row is 1,1,5

Step 7: Divide by 5 

• 1

• 1

• 5 ÷ 5 = 1

Now the row is 1,1,1

Step 8: Multiply  all the divisors used

LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360

 

Solved Examples

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: 

• 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

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.

 

FAQs on Prime Factorization of HCF and LCM

1. What is the formula for HCF and LCM?

Ans: 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 the prime factorisation method?

Ans: 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?

Ans: 

• 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 that divides 245 and 1037, leaving a remainder of 5 in each case.

4. How do you factor HCF and LCM?

Ans: 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 out 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?

Ans: HCF=6,LCM=504. Step-by-step video, text & image solution for using the prime factorization method to find the HCF and LCM of 72, 126, and 168.