Using prime factorisation find HCF and LCM of 18,45 and 60 check if HCF x LCM = product of the number

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 18, 45, and 60 using prime factorization, follow these steps:

### Step 1: Prime Factorization of Each Number
- **For 18**:
- Divide by 2: \( 18 \div 2 = 9 \)
- Divide by 3: \( 9 \div 3 = 3 \)
- Divide by 3: \( 3 \div 3 = 1 \)
- Prime factorization: \( 18 = 2^1 \times 3^2 \)

- **For 45**:
- Divide by 3: \( 45 \div 3 = 15 \)
- Divide by 3: \( 15 \div 3 = 5 \)
- Divide by 5: \( 5 \div 5 = 1 \)
- Prime factorization: \( 45 = 3^2 \times 5^1 \)

- **For 60**:
- Divide by 2: \( 60 \div 2 = 30 \)
- Divide by 2: \( 30 \div 2 = 15 \)
- Divide by 3: \( 15 \div 3 = 5 \)
- Divide by 5: \( 5 \div 5 = 1 \)
- Prime factorization: \( 60 = 2^2 \times 3^1 \times 5^1 \)

### Step 2: Determine HCF
To find the HCF, take the lowest power of all prime factors present in all numbers:
- For \( 2 \): The lowest power is \( 2^0 \) (not present in 45).
- For \( 3 \): The lowest power is \( 3^1 \) (present in all).
- For \( 5 \): The lowest power is \( 5^0 \) (not present in 18).

Thus, the HCF is:
\[ \text{HCF} = 2^0 \times 3^1 \times 5^0 = 3 \]

### Step 3: Determine LCM
To find the LCM, take the highest power of all prime factors present in any of the numbers:
- For \( 2 \): The highest power is \( 2^2 \) (from 60).
- For \( 3 \): The highest power is \( 3^2 \) (from 18 and 45).
- For \( 5 \): The highest power is \( 5^1 \) (from 45 and 60).

Thus, the LCM is:
\[ \text{LCM} = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180 \]

### Step 4: Verify HCF × LCM = Product of the Numbers
Now, we check if:
\[ \text{HCF} \times \text{LCM} = 3 \times 180 = 540 \]

Calculating the product of the numbers:
\[ 18 \times 45 \times 60 = 8100 \]

### Conclusion
Since \( 540 \) does not equal \( 8100 \), we can conclude that:
\[ \text{HCF} \times \text{LCM} \neq \text{Product of the numbers} \]

### Summary
- HCF of 18, 45, and 60 is **3**.
- LCM of 18, 45, and 60 is **180**.
- The product of the numbers is **8100**, which does not equal \( \text{HCF} \times \text{LCM} \).