Find the HCF and LCM of 32 and 48.

To find the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM) of the numbers 32 and 48, we can use a systematic approach. Let's break it down step by step.

Finding the HCF

The HCF, also known as the greatest common divisor (GCD), is the largest number that divides both numbers without leaving a remainder. One effective method to find the HCF is to use the prime factorization of each number.

Step 1: Prime Factorization

First, we need to find the prime factors of both 32 and 48.

• 32: The prime factorization of 32 is 2 × 2 × 2 × 2 × 2, which can be written as 2⁵.
• 48: The prime factorization of 48 is 2 × 2 × 2 × 2 × 3, which can be expressed as 2⁴ × 3.

Step 2: Identify Common Factors

Next, we identify the common prime factors. Both numbers share the prime factor 2. The lowest power of 2 in the factorizations is 2⁴.

Step 3: Calculate the HCF

Thus, the HCF of 32 and 48 is:

HCF = 2⁴ = 16

Calculating the LCM

The LCM is the smallest number that both 32 and 48 can divide into without a remainder. To find the LCM, we can also use the prime factorization method.

Step 1: Use Prime Factorization Again

We already have the prime factorizations:

• 32 = 2⁵
• 48 = 2⁴ × 3

Step 2: Take the Highest Powers

For the LCM, we take the highest power of each prime factor present in either number:

• For the prime factor 2, the highest power is 2⁵.
• For the prime factor 3, the highest power is 3¹.

Step 3: Calculate the LCM

Now, we multiply these together to find the LCM:

LCM = 2⁵ × 3¹ = 32 × 3 = 96

Summary of Results

In conclusion, for the numbers 32 and 48:

• HCF = 16
• LCM = 96

This method of using prime factorization is quite effective and can be applied to any pair of numbers to find their HCF and LCM. If you have any more questions or need further clarification, feel free to ask!