Find the HCF and LCM of 12, 72, and 120 by Prime Factorization method.

To find the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of the numbers 12, 72, and 120 using the prime factorization method, we first need to break each number down into its prime factors. This will allow us to identify the common factors and the multiples effectively.

Step 1: Prime Factorization of Each Number

Let's start with the prime factorization of each number:

• 12: The prime factors of 12 can be found by dividing by the smallest prime numbers:
• 12 ÷ 2 = 6
• 6 ÷ 2 = 3
• 3 ÷ 3 = 1

So, the prime factorization of 12 is 2² × 3¹.

• 72: Now, let's factor 72:
• 72 ÷ 2 = 36
• 36 ÷ 2 = 18
• 18 ÷ 2 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

The prime factorization of 72 is 2³ × 3².

• 120: Finally, we factor 120:
• 120 ÷ 2 = 60
• 60 ÷ 2 = 30
• 30 ÷ 2 = 15
• 15 ÷ 3 = 5
• 5 ÷ 5 = 1

The prime factorization of 120 is 2³ × 3¹ × 5¹.

Step 2: Finding the HCF

The HCF is found by taking the lowest power of all prime factors common to each number:

• For the prime factor 2: The lowest power is 2² (from 12).
• For the prime factor 3: The lowest power is 3¹ (from both 12 and 120).
• Prime factor 5 is not common to all three numbers.

Thus, the HCF is:

HCF = 2² × 3¹ = 4 × 3 = 12

Step 3: Finding the LCM

The LCM is determined by taking the highest power of all prime factors present in any of the numbers:

• For the prime factor 2: The highest power is 2³ (from both 72 and 120).
• For the prime factor 3: The highest power is 3² (from 72).
• For the prime factor 5: The highest power is 5¹ (from 120).

Therefore, the LCM is:

LCM = 2³ × 3² × 5¹ = 8 × 9 × 5

Calculating this step-by-step:

• 8 × 9 = 72
• 72 × 5 = 360

So, the LCM is 360.

Final Results

In summary, for the numbers 12, 72, and 120:

• HCF: 12
• LCM: 360

This method of prime factorization not only helps in finding the HCF and LCM but also deepens your understanding of the relationships between numbers. If you have any further questions or need clarification on any step, feel free to ask!