If the sum of two numbers is 352 and their H.C.F. is 32, then the possible pairs of such numbers are :

Question

Accepted Answer

To determine the number of possible pairs of numbers whose sum is 352 and whose highest common factor (H.C.F.) is 32, we need to consider the properties of numbers and their divisors.

### Explanation:

1. **Understanding the Problem:**
   - We are given that the sum of two numbers is 352.
   - The highest common factor (H.C.F.) of these two numbers is 32.

2. **Formulating the Numbers:**
   - Let the two numbers be $32a$ and $32b$, where $a$ and $b$ are coprime integers (since the H.C.F. is 32, the numbers must be multiples of 32).
   - The equation based on the sum is: 
     $$
     32a + 32b = 352
     $$
   - Simplifying this equation, we get:
     $$
     a + b = \frac{352}{32} = 11
     $$

3. **Finding Pairs of Coprime Numbers:**
   - We need to find pairs $(a, b)$ such that $a + b = 11$ and $a$ and $b$ are coprime.
   - The possible pairs $(a, b)$ that satisfy $a + b = 11$ are:
     - (1, 10)
     - (2, 9)
     - (3, 8)
     - (4, 7)
     - (5, 6)

4. **Checking Coprimeness:**
   - For each pair, check if $a$ and $b$ are coprime:
     - (1, 10): Coprime
     - (2, 9): Coprime
     - (3, 8): Coprime
     - (4, 7): Coprime
     - (5, 6): Coprime

5. **Conclusion:**
   - All five pairs are coprime, meaning they satisfy the condition that their H.C.F. is 1.
   - Therefore, when multiplied by 32, each pair results in two numbers whose H.C.F. is 32 and whose sum is 352.

### Why Other Options are Incorrect:

- **Option 2 (Six pairs):** There are no additional pairs that satisfy both the sum and coprimeness conditions beyond the five identified.
- **Option 3 (Four pairs):** This option is incorrect because we have identified five valid pairs.
- **Option 4 (Three pairs):** This option is incorrect because it underestimates the number of valid pairs, which is five.

Thus, the correct answer is Option 1, as there are exactly five pairs of numbers that meet the given conditions.