Find LCM and HCF of 510 and 92; Verify LCM and HCF is Equal to the Product of Two Numbers

To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the numbers 510 and 92, we first need to factor both numbers into their prime factors.

Step 1: Prime Factorization

For 510:

• 510 is divisible by 2: 510 ÷ 2 = 255
• 255 is divisible by 3: 255 ÷ 3 = 85
• 85 is divisible by 5: 85 ÷ 5 = 17
• 17 is a prime number.

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

For 92:

• 92 is divisible by 2: 92 ÷ 2 = 46
• 46 is divisible by 2: 46 ÷ 2 = 23
• 23 is a prime number.

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

Step 2: Finding HCF

The HCF is found by taking the lowest power of all common prime factors. The common factor between 510 and 92 is 2.

• HCF = 2¹ = 2

Step 3: Finding LCM

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

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

Now, taking the highest powers:

• LCM = 2² × 3¹ × 5¹ × 17¹ × 23¹

Calculating this, we find:

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

So, the LCM of 510 and 92 is: 23460.

Step 4: Verification

Now, we’ll verify that the product of the two numbers is equal to the product of their LCM and HCF:

• Product of the two numbers: 510 × 92 = 46920
• Product of LCM and HCF: 23460 × 2 = 46920

Since both products give us 46920, we can conclude that our calculations are correct.

Therefore, the LCM of 510 and 92 is 23460, and the HCF is 2.