Find HCF and LCM of 510 and 92 by applying the prime factorization method.

To find the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of the numbers 510 and 92 using the prime factorization method, we first need to break down each number into its prime factors. This process will help us identify the common factors and the multiples of the two numbers effectively.

Step 1: Prime Factorization of Each Number

Let's start with 510. We can divide it by the smallest prime number, which is 2:

• 510 ÷ 2 = 255

Next, we factor 255. The smallest prime number that divides 255 is 3:

• 255 ÷ 3 = 85

Now, we factor 85. The smallest prime number that divides 85 is 5:

• 85 ÷ 5 = 17

Finally, 17 is a prime number itself. Thus, the prime factorization of 510 is:

• 510 = 2 × 3 × 5 × 17

Now, let's factor 92. We start again with the smallest prime number, 2:

• 92 ÷ 2 = 46
• 46 ÷ 2 = 23

Since 23 is also a prime number, the prime factorization of 92 is:

• 92 = 2² × 23

Step 2: Finding the HCF

The HCF is found by taking the product of the lowest powers of all prime factors common to both numbers. From our factorizations:

• Prime factors of 510: 2, 3, 5, 17
• Prime factors of 92: 2², 23

The only common prime factor is 2. The lowest power of 2 in both factorizations is 2¹. Therefore, the HCF is:

• HCF = 2¹ = 2

Step 3: Finding the LCM

The LCM is calculated by taking the product of the highest powers of all prime factors present in either number. From our previous factorizations, we have:

• From 510: 2¹, 3¹, 5¹, 17¹
• From 92: 2², 23¹

Now, we take the highest powers:

• 2² (from 92)
• 3¹ (from 510)
• 5¹ (from 510)
• 17¹ (from 510)
• 23¹ (from 92)

Now, we multiply these together to find the LCM:

• LCM = 2² × 3¹ × 5¹ × 17¹ × 23¹

Calculating this step-by-step:

• 2² = 4
• 4 × 3 = 12
• 12 × 5 = 60
• 60 × 17 = 1020
• 1020 × 23 = 23460

Thus, the LCM of 510 and 92 is:

• LCM = 23460

Final Results

To summarize:

• HCF of 510 and 92: 2
• LCM of 510 and 92: 23460

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