The sum of two numbers is 232 and their HCF is 29. What is the number of such pairs of numbers satisfying the above condition ?

Options

A. One
B. Two
C. Four
D. None of theser

Answer

To solve the problem step by step, we need to find pairs of numbers whose sum is 232 and whose highest common factor (HCF) is 29.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know the sum of two numbers is 232 and their HCF is 29. Let's denote the two numbers as \( x \) and \( y \).

2. **Expressing the Numbers in Terms of HCF**:
Since the HCF of \( x \) and \( y \) is 29, we can express the numbers as:
\[ x = 29a \quad \text{and} \quad y = 29b \]
where \( a \) and \( b \) are co-prime integers (they have no common factors other than 1).

3. **Setting Up the Equation**:
According to the problem, the sum of the two numbers is:
\[ x + y = 232 \]
Substituting the expressions for \( x \) and \( y \):
\[ 29a + 29b = 232 \]

4. **Simplifying the Equation**:
We can factor out 29 from the left side:
\[ 29(a + b) = 232 \]
Dividing both sides by 29 gives:
\[ a + b = \frac{232}{29} = 8 \]

5. **Finding Co-prime Pairs**:
Now we need to find pairs of positive integers \( (a, b) \) such that:
\[ a + b = 8 \]
and \( a \) and \( b \) are co-prime.

The possible pairs \( (a, b) \) that satisfy \( a + b = 8 \) are:
- \( (1, 7) \)
- \( (2, 6) \)
- \( (3, 5) \)
- \( (4, 4) \)

6. **Checking for Co-primality**:
- For \( (1, 7) \): GCD(1, 7) = 1 (co-prime)
- For \( (2, 6) \): GCD(2, 6) = 2 (not co-prime)
- For \( (3, 5) \): GCD(3, 5) = 1 (co-prime)
- For \( (4, 4) \): GCD(4, 4) = 4 (not co-prime)

Thus, the valid pairs are \( (1, 7) \) and \( (3, 5) \).

7. **Counting the Valid Pairs**:
We found two valid pairs of \( (a, b) \) that are co-prime:
- \( (1, 7) \)
- \( (3, 5) \)

Therefore, the number of such pairs of numbers \( (x, y) \) that satisfy the given conditions is **2**.

### Final Answer:
The number of pairs of numbers satisfying the conditions is **2**.

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