The LCM and HCF of the numbers 28 and 42 are in the ratio:

Options

A. `6:1`
B. `2:3`
C. `3:2`
D. `7:2`

Answer

To find the ratio of the LCM and HCF of the numbers 28 and 42, we will follow these steps:

### Step 1: Find the LCM of 28 and 42
To find the LCM (Least Common Multiple), we can use the prime factorization method.

1. **Prime Factorization of 28:**
- 28 can be factored as:
- \( 28 = 7 \times 4 = 7 \times 2^2 \)
- So, the prime factorization of 28 is \( 2^2 \times 7^1 \).

2. **Prime Factorization of 42:**
- 42 can be factored as:
- \( 42 = 7 \times 6 = 7 \times 2 \times 3 \)
- So, the prime factorization of 42 is \( 2^1 \times 3^1 \times 7^1 \).

3. **Finding LCM:**
- To find the LCM, take the highest power of each prime number from the factorizations:
- From 28: \( 2^2 \) and \( 7^1 \)
- From 42: \( 2^1 \), \( 3^1 \), and \( 7^1 \)
- Therefore, LCM = \( 2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7 \).
- Calculating this gives:
- \( 4 \times 3 = 12 \)
- \( 12 \times 7 = 84 \)
- Thus, LCM(28, 42) = 84.

### Step 2: Find the HCF of 28 and 42
To find the HCF (Highest Common Factor), we can also use the prime factorization method.

1. **Using the prime factorizations from above:**
- For 28: \( 2^2 \times 7^1 \)
- For 42: \( 2^1 \times 3^1 \times 7^1 \)

2. **Finding HCF:**
- Take the lowest power of each common prime factor:
- For \( 2 \): minimum power is \( 2^1 \)
- For \( 7 \): minimum power is \( 7^1 \)
- Therefore, HCF = \( 2^1 \times 7^1 = 2 \times 7 = 14 \).
- Thus, HCF(28, 42) = 14.

### Step 3: Calculate the Ratio of LCM to HCF
Now we can find the ratio of LCM to HCF:

\[ \text{Ratio} = \frac{\text{LCM}}{\text{HCF}} = \frac{84}{14} \]

Calculating this gives:

\[ \frac{84}{14} = 6 \]

So, the ratio of LCM to HCF is \( 6:1 \).

### Final Answer
The ratio of the LCM and HCF of the numbers 28 and 42 is \( 6:1 \).

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