The L.C.M. of two numbers is 1020 and their H.C.F. is 34 the possible pair of numbers is :

Options

A. 255, 34
B. 102, 204
C. 204, 170
D. None of these

Answer

To find the possible pair of numbers given their L.C.M. and H.C.F., we can use the relationship between these values. The formula we will use is:

\[ \text{L.C.M.} \times \text{H.C.F.} = \text{Product of the two numbers} \]

### Step-by-Step Solution:

1. **Identify the given values**:
- L.C.M. = 1020
- H.C.F. = 34

2. **Use the formula to find the product of the two numbers**:
\[ \text{Product of the two numbers} = \text{L.C.M.} \times \text{H.C.F.} \]
\[ = 1020 \times 34 \]

3. **Calculate the product**:
\[ 1020 \times 34 = 34680 \]
So, the product of the two numbers is 34680.

4. **Let the two numbers be \( a \) and \( b \)**. We know:
- \( a \times b = 34680 \)
- \( \text{H.C.F.}(a, b) = 34 \)

5. **Express the two numbers in terms of their H.C.F.**:
- Let \( a = 34m \) and \( b = 34n \), where \( m \) and \( n \) are co-prime integers (since H.C.F. is 34).

6. **Substitute into the product equation**:
\[ (34m) \times (34n) = 34680 \]
\[ 34^2 \times (m \times n) = 34680 \]
\[ 1156 \times (m \times n) = 34680 \]

7. **Solve for \( m \times n \)**:
\[ m \times n = \frac{34680}{1156} \]
\[ m \times n = 30 \]

8. **Find pairs of \( (m, n) \) that multiply to 30**:
The pairs of integers \( (m, n) \) that satisfy \( m \times n = 30 \) are:
- (1, 30)
- (2, 15)
- (3, 10)
- (5, 6)

9. **Calculate the corresponding pairs of \( (a, b) \)**:
- For \( (1, 30) \): \( a = 34 \times 1 = 34, b = 34 \times 30 = 1020 \) → Pair: (34, 1020)
- For \( (2, 15) \): \( a = 34 \times 2 = 68, b = 34 \times 15 = 510 \) → Pair: (68, 510)
- For \( (3, 10) \): \( a = 34 \times 3 = 102, b = 34 \times 10 = 340 \) → Pair: (102, 340)
- For \( (5, 6) \): \( a = 34 \times 5 = 170, b = 34 \times 6 = 204 \) → Pair: (170, 204)

10. **Conclusion**:
The possible pairs of numbers are (34, 1020), (68, 510), (102, 340), and (170, 204). The pair that satisfies the condition of having L.C.M. = 1020 and H.C.F. = 34 is (170, 204).