The LCM and the 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 (Least Common Multiple) and HCF (Highest Common Factor) of the numbers 28 and 42, we will follow these steps:

### Step 1: Find the HCF of 28 and 42
To find the HCF, we can use the prime factorization method.

- **Prime factorization of 28**:
- 28 = 2 × 2 × 7 = \(2^2 \times 7^1\)

- **Prime factorization of 42**:
- 42 = 2 × 3 × 7 = \(2^1 \times 3^1 \times 7^1\)

Now, we take the lowest power of each common prime factor:
- For 2: min(2, 1) = 1 → \(2^1\)
- For 3: min(0, 1) = 0 → \(3^0\) (not included)
- For 7: min(1, 1) = 1 → \(7^1\)

Thus, the HCF = \(2^1 \times 7^1 = 2 \times 7 = 14\).

### Step 2: Find the LCM of 28 and 42
To find the LCM, we take the highest power of each prime factor present in both numbers:

- For 2: max(2, 1) = 2 → \(2^2\)
- For 3: max(0, 1) = 1 → \(3^1\)
- For 7: max(1, 1) = 1 → \(7^1\)

Thus, the LCM = \(2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7\).

Calculating this step-by-step:
- \(4 \times 3 = 12\)
- \(12 \times 7 = 84\)

So, the LCM = 84.

### Step 3: Find 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:
\[ \frac{84}{14} = 6 \]

### Final Result
The ratio of the LCM and HCF of the numbers 28 and 42 is 6:1.

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