The H.C.F. of two numbers is 43 and their sum is 430. Total number of distinct pairs of two such numbers is :

Options

A. 5
B. 2
C. 6
D. data insufficient

Answer

To solve the problem, we need to find the total number of distinct pairs of two numbers given that their H.C.F. is 43 and their sum is 430.

### Step-by-Step Solution:

1. **Understand the relationship between the numbers:**
Let the two numbers be \( a \) and \( b \). Given that their H.C.F. is 43, we can express the numbers as:
\[ a = 43x \quad \text{and} \quad b = 43y \]
where \( x \) and \( y \) are co-prime integers (i.e., they have no common factors other than 1).

2. **Set up the equation for their sum:**
According to the problem, the sum of the two numbers is:
\[ a + b = 430 \]
Substituting the expressions for \( a \) and \( b \):
\[ 43x + 43y = 430 \]

3. **Simplify the equation:**
We can factor out 43 from the left side:
\[ 43(x + y) = 430 \]
Dividing both sides by 43 gives:
\[ x + y = 10 \]

4. **Find pairs of co-prime integers:**
Now, we need to find pairs of integers \( (x, y) \) such that \( x + y = 10 \) and \( x \) and \( y \) are co-prime. The possible pairs of integers that sum to 10 are:
- (1, 9)
- (2, 8)
- (3, 7)
- (4, 6)
- (5, 5)

5. **Check for co-primality:**
Now we check which of these pairs are co-prime:
- (1, 9): Co-prime
- (2, 8): Not co-prime (common factor is 2)
- (3, 7): Co-prime
- (4, 6): Not co-prime (common factor is 2)
- (5, 5): Not co-prime (same number)

The valid co-prime pairs are:
- (1, 9)
- (3, 7)

6. **Count the distinct pairs:**
We have found 2 distinct pairs of \( (x, y) \):
- (1, 9)
- (3, 7)

Therefore, the total number of distinct pairs of the two numbers \( a \) and \( b \) is 2.

### Final Answer:
The total number of distinct pairs of two such numbers is **2**.