If the LCM of two numbers is 70 and their HCF is 2, find the numbers.

Options

A. 2, 35
B. 6, 70
C. 4, 70
D. 14, 10

Answer

To find the two numbers given their LCM (Least Common Multiple) and HCF (Highest Common Factor), we can use the relationship between these values.

### Step-by-Step Solution:

1. **Understand the relationship between LCM, HCF, and the two numbers:**
The relationship is given by the formula:
\[ \text{LCM} \times \text{HCF} = \text{Product of the two numbers} \]

2. **Substitute the given values:**
Here, we know:
- LCM = 70
- HCF = 2

Substituting these values into the formula gives us:
\[ 70 \times 2 = \text{Product of the two numbers} \]
\[ 140 = \text{Product of the two numbers} \]

3. **Let the two numbers be \( a \) and \( b \):**
We can express this as:
\[ a \times b = 140 \]

4. **Find pairs of factors of 140:**
We need to find pairs of numbers \( (a, b) \) such that their product is 140. The pairs of factors of 140 are:
- (1, 140)
- (2, 70)
- (4, 35)
- (5, 28)
- (7, 20)
- (10, 14)

5. **Check which pair has HCF of 2:**
Now we need to check which of these pairs has an HCF of 2:
- For (1, 140), HCF is 1.
- For (2, 70), HCF is 2.
- For (4, 35), HCF is 7.
- For (5, 28), HCF is 1.
- For (7, 20), HCF is 1.
- For (10, 14), HCF is 2.

The pairs that have an HCF of 2 are (2, 70) and (10, 14).

6. **Verify LCM for valid pairs:**
We need to check the LCM for the pairs (2, 70) and (10, 14):
- LCM(2, 70) = 70 (valid)
- LCM(10, 14) = 70 (valid)

7. **Conclusion:**
The two pairs of numbers that satisfy the conditions are (2, 70) and (10, 14). Thus, the numbers can be either:
- 2 and 70
- 10 and 14