Find the LCM and HCF of the following pair of integers and verify that LCM ` xx`HCF = Product of the two numbers
336 and 54

Answer

To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the integers 336 and 54, and verify that LCM × HCF = Product of the two numbers, we can follow these steps:

### Step 1: Prime Factorization of 336
To find the prime factorization of 336, we divide it by the smallest prime numbers until we reach 1.

- 336 ÷ 2 = 168
- 168 ÷ 2 = 84
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1

So, the prime factorization of 336 is:
\[ 336 = 2^4 \times 3^1 \times 7^1 \]

### Step 2: Prime Factorization of 54
Now, we perform the same for 54.

- 54 ÷ 2 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 54 is:
\[ 54 = 2^1 \times 3^3 \]

### Step 3: Finding HCF
The HCF is found by taking the lowest power of all prime factors common to both numbers.

- For 2: The lowest power is \(2^1\)
- For 3: The lowest power is \(3^1\)

Thus, the HCF is:
\[ HCF = 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Step 4: Finding LCM
The LCM is found by taking the highest power of all prime factors present in either number.

- For 2: The highest power is \(2^4\)
- For 3: The highest power is \(3^3\)
- For 7: The highest power is \(7^1\)

Thus, the LCM is:
\[ LCM = 2^4 \times 3^3 \times 7^1 \]

Calculating this:
- \(2^4 = 16\)
- \(3^3 = 27\)
- \(7^1 = 7\)

Now, multiply these together:
\[ LCM = 16 \times 27 \times 7 \]

Calculating step by step:
- \(16 \times 27 = 432\)
- \(432 \times 7 = 3024\)

So, the LCM is:
\[ LCM = 3024 \]

### Step 5: Verification
Now, we need to verify that:
\[ LCM \times HCF = 336 \times 54 \]

Calculating the product of LCM and HCF:
\[ LCM \times HCF = 3024 \times 6 = 18144 \]

Now calculating the product of the two numbers:
\[ 336 \times 54 = 18144 \]

Since both products are equal:
\[ LCM \times HCF = 18144 \]
\[ 336 \times 54 = 18144 \]

### Conclusion
Thus, we have found:
- HCF = 6
- LCM = 3024
And verified that:
\[ LCM \times HCF = 336 \times 54 \]