Find the ratio of LCM and HCF of 28 and 42.

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 prime factorization of both numbers.
- **For 28**:
- 28 can be divided by 2: \( 28 \div 2 = 14 \)
- 14 can be divided by 2: \( 14 \div 2 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 28 is \( 2^2 \times 7^1 \).

- **For 42**:
- 42 can be divided by 2: \( 42 \div 2 = 21 \)
- 21 can be divided by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 42 is \( 2^1 \times 3^1 \times 7^1 \).

### Step 2: Find the LCM of 28 and 42.
- The LCM is found by taking the highest power of each prime number present in the factorizations.
- For the prime number 2: the highest power is \( 2^2 \) (from 28).
- For the prime number 3: the highest power is \( 3^1 \) (from 42).
- For the prime number 7: the highest power is \( 7^1 \) (common in both).

- Therefore, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7 = 84. \]

### Step 3: Find the HCF of 28 and 42.
- The HCF is found by taking the lowest power of each prime number common to both factorizations.
- For the prime number 2: the lowest power is \( 2^1 \) (from 42).
- For the prime number 7: the lowest power is \( 7^1 \) (common in both).

- Therefore, the HCF is:
\[ HCF = 2^1 \times 7^1 = 2 \times 7 = 14. \]

### Step 4: Find the ratio of LCM to HCF.
- Now we can find the ratio of LCM to HCF:
\[ \text{Ratio} = \frac{LCM}{HCF} = \frac{84}{14} = 6. \]

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

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