Find the HCF and LCM of the following pairs of integers and verify that LCM × HCF = product of the two numbers.(i) 510 and 92

To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of 510 and 92, we will follow these steps:

Step 1: Find the prime factorization of both numbers
Prime factorization of 510:
510 ÷ 2 = 255 (since 510 is even) 255 ÷ 3 = 85 (sum of digits of 255 is 12, which is divisible by 3) 85 ÷ 5 = 17 (last digit is 5, so divisible by 5) 17 is a prime number.

So, the prime factorization of 510 is: 510 = 2 × 3 × 5 × 17

Prime factorization of 92:
92 ÷ 2 = 46 (since 92 is even) 46 ÷ 2 = 23 (since 46 is even) 23 is a prime number.

So, the prime factorization of 92 is: 92 = 2² × 23

Step 2: Find the HCF (Highest Common Factor)
The HCF is found by taking the lowest power of all the common prime factors between the two numbers.

Common prime factor between 510 and 92 is 2.

The lowest power of 2 common in both factorizations is 2¹.
Thus, the HCF of 510 and 92 is: HCF = 2

Step 3: Find the LCM (Least Common Multiple)
The LCM is found by taking the highest power of each prime factor from both numbers.

For 2, the highest power is 2² (from 92).
For 3, the highest power is 3¹ (from 510).
For 5, the highest power is 5¹ (from 510).
For 17, the highest power is 17¹ (from 510).
For 23, the highest power is 23¹ (from 92).
Thus, the LCM of 510 and 92 is: LCM = 2² × 3¹ × 5¹ × 17¹ × 23¹ LCM = 4 × 3 × 5 × 17 × 23 LCM = 4 × 3 = 12 12 × 5 = 60 60 × 17 = 1020 1020 × 23 = 23460

So, the LCM of 510 and 92 is 23460.

Step 4: Verify that LCM × HCF = product of the two numbers
Now, let's verify the relationship: LCM × HCF = 23460 × 2 = 46920

The product of 510 and 92 is: 510 × 92 = 46920

Since both values are the same, we can confirm that: LCM × HCF = product of the two numbers.

Final Answer:
HCF of 510 and 92 = 2 LCM of 510 and 92 = 23460 Verification: LCM × HCF = 46920, which is the product of 510 and 92.