How do you find the LCM and HCF of 510 and 92, and how can we verify that their product equals the LCM and HCF?

Finding the LCM and HCF of 510 and 92

Step 1: Prime Factorization

To find the Least Common Multiple (LCM) and Highest Common Factor (HCF), we can start by finding the prime factorization of both numbers:

Prime Factorization of 510:

• 510 can be divided by 2: 510 = 2 × 255
• 255 can be divided by 3: 255 = 3 × 85
• 85 can be divided by 5: 85 = 5 × 17
• 17 is a prime number.

So, the complete prime factorization of 510 is:

510 = 2 × 3 × 5 × 17

Prime Factorization of 92:

• 92 can be divided by 2: 92 = 2 × 46
• 46 can be divided by 2: 46 = 2 × 23
• 23 is a prime number.

Thus, the prime factorization of 92 is:

92 = 2² × 23

Step 2: Finding HCF

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

• Common prime factor: 2

Thus, the HCF is:

HCF = 2¹ = 2

Step 3: Finding LCM

The LCM is found by taking the highest power of all prime factors present in either number:

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

Thus, the LCM is:

LCM = 2² × 3 × 5 × 17 × 23 = 23460

Step 4: Verification of LCM and HCF

To verify that the product of the two numbers is equal to the product of the LCM and HCF, we can perform the following calculation:

Product of the two numbers:

510 × 92 = 46920

Product of LCM and HCF:

23460 × 2 = 46920

Since both products are equal (46920 = 46920), we have verified the relationship:

Product of LCM and HCF = Product of the numbers

Conclusion

In summary, for the numbers 510 and 92, the HCF is 2, the LCM is 23460, and we have confirmed that:

LCM × HCF = Product of the two numbers