Calculate the ratio between the LCM and HCF of the numbers 5, 15 and 20.

Options

A. `5 : 3`
B. `7 : 2`
C. `9 : 4`
D. `12 : 1`

Answer

To calculate the ratio between the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the numbers 5, 15, and 20, we will follow these steps:

### Step 1: Find the LCM of 5, 15, and 20
To find the LCM, we can use the prime factorization method.

- **Prime Factorization:**
- \(5 = 5^1\)
- \(15 = 3^1 \times 5^1\)
- \(20 = 2^2 \times 5^1\)

- **LCM Calculation:**
The LCM is found by taking the highest power of each prime number present in the factorizations:
- For \(2\), the highest power is \(2^2\) (from 20).
- For \(3\), the highest power is \(3^1\) (from 15).
- For \(5\), the highest power is \(5^1\) (common in all).

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \]

### Step 2: Find the HCF of 5, 15, and 20
To find the HCF, we again use the prime factorization.

- **HCF Calculation:**
The HCF is found by taking the lowest power of each common prime number:
- For \(5\), the lowest power is \(5^1\) (common in all).
- For \(3\) and \(2\), they are not common in all three numbers.

Thus, the HCF is:
\[ HCF = 5^1 = 5 \]

### Step 3: Calculate the Ratio of LCM to HCF
Now, we can calculate the ratio of LCM to HCF:
\[ \text{Ratio} = \frac{LCM}{HCF} = \frac{60}{5} = 12 \]

### Final Answer
The ratio between the LCM and HCF of the numbers 5, 15, and 20 is:
\[ \text{Ratio} = 12 : 1 \]

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