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 that the L.C.M. is 1020 and the H.C.F. is 34, we can use the relationship between L.C.M., H.C.F., and the two numbers.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The relationship between the two numbers (let's call them \( a \) and \( b \)), their L.C.M. (Least Common Multiple), and H.C.F. (Highest Common Factor) is given by:
\[ a \times b = \text{L.C.M.} \times \text{H.C.F.} \]

2. **Substitute the Known Values**:
We know:
- L.C.M. = 1020
- H.C.F. = 34
Therefore, we can write:
\[ a \times b = 1020 \times 34 \]

3. **Calculate the Product**:
Now, calculate \( 1020 \times 34 \):
\[ 1020 \times 34 = 34680 \]
So, we have:
\[ a \times b = 34680 \]

4. **Express the Numbers in Terms of H.C.F.**:
Since the H.C.F. of the two numbers is 34, we can express the two numbers as:
\[ a = 34x \quad \text{and} \quad b = 34y \]
where \( x \) and \( y \) are coprime integers (i.e., their H.C.F. is 1).

5. **Substitute into the Product Equation**:
Substitute \( a \) and \( b \) into the product equation:
\[ (34x) \times (34y) = 34680 \]
This simplifies to:
\[ 1156xy = 34680 \]

6. **Solve for \( xy \)**:
Now, divide both sides by 1156:
\[ xy = \frac{34680}{1156} \]
Calculate \( \frac{34680}{1156} \):
\[ xy = 30 \]

7. **Find Pairs of \( (x, y) \)**:
Now we need to find pairs of integers \( (x, y) \) such that \( xy = 30 \). The pairs are:
- (1, 30)
- (2, 15)
- (3, 10)
- (5, 6)
- (6, 5)
- (10, 3)
- (15, 2)
- (30, 1)

8. **Calculate the Corresponding Values of \( a \) and \( b \)**:
For each pair \( (x, y) \), calculate \( a \) and \( b \):
- For \( (1, 30) \): \( a = 34 \times 1 = 34 \), \( b = 34 \times 30 = 1020 \)
- For \( (2, 15) \): \( a = 34 \times 2 = 68 \), \( b = 34 \times 15 = 510 \)
- For \( (3, 10) \): \( a = 34 \times 3 = 102 \), \( b = 34 \times 10 = 340 \)
- For \( (5, 6) \): \( a = 34 \times 5 = 170 \), \( b = 34 \times 6 = 204 \)

9. **List the Possible Pairs**:
The possible pairs of numbers are:
- (34, 1020)
- (68, 510)
- (102, 340)
- (170, 204)