Find the HCF and LCM of 24 and 36 by prime factorization method.

To find the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of the numbers 24 and 36 using the prime factorization method, we first need to break down each number into its prime factors. This method is effective because it allows us to see the building blocks of each number clearly.

Step 1: Prime Factorization of Each Number

Let's start with 24. We can divide it by the smallest prime number, which is 2:

• 24 ÷ 2 = 12
• 12 ÷ 2 = 6
• 6 ÷ 2 = 3
• 3 ÷ 3 = 1

So, the prime factorization of 24 is:

24 = 2^3 × 3^1

Now, let's factor 36 in a similar way:

• 36 ÷ 2 = 18
• 18 ÷ 2 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

The prime factorization of 36 is:

36 = 2^2 × 3^2

Step 2: Finding the HCF

The HCF is found by taking the lowest power of each common prime factor from the factorizations:

• For the prime factor 2: The lowest power is 2^2 (from 36).
• For the prime factor 3: The lowest power is 3^1 (from 24).

Now, we multiply these together to find the HCF:

HCF = 2^2 × 3^1 = 4 × 3 = 12

Step 3: Calculating the LCM

The LCM is determined by taking the highest power of each prime factor present in either number:

• For the prime factor 2: The highest power is 2^3 (from 24).
• For the prime factor 3: The highest power is 3^2 (from 36).

Now, we multiply these together to find the LCM:

LCM = 2^3 × 3^2 = 8 × 9 = 72

Summary of Results

To summarize our findings:

• HCF of 24 and 36 is 12
• LCM of 24 and 36 is 72

This method of prime factorization not only helps in finding the HCF and LCM but also reinforces the understanding of how numbers are constructed from their prime factors. By recognizing these factors, you can easily tackle similar problems in the future!