The sum of two numbers is 84 and their HCF is 12. Total number of such pairs is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem step by step, we need to find the total number of pairs of two numbers that sum to 84 and have an HCF of 12.

### Step-by-Step Solution:

1. **Understanding the Problem**: We know that the sum of two numbers is 84 and their highest common factor (HCF) is 12. This means both numbers can be expressed as multiples of 12.

2. **Expressing the Numbers**: Let the two numbers be \( 12x \) and \( 12y \), where \( x \) and \( y \) are integers. Since the HCF is 12, \( x \) and \( y \) must be coprime (i.e., their HCF is 1).

3. **Setting Up the Equation**: According to the problem, we have:
\[ 12x + 12y = 84 \]
We can factor out 12:
\[ 12(x + y) = 84 \]
Dividing both sides by 12 gives:
\[ x + y = 7 \]

4. **Finding Pairs of \( x \) and \( y \)**: Now we need to find pairs of integers \( (x, y) \) such that \( x + y = 7 \) and \( \text{HCF}(x, y) = 1 \).

5. **Listing Possible Pairs**: The possible pairs of \( (x, y) \) that sum to 7 are:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)

6. **Checking for Coprimality**:
- For (1, 6): HCF(1, 6) = 1 (valid)
- For (2, 5): HCF(2, 5) = 1 (valid)
- For (3, 4): HCF(3, 4) = 1 (valid)
- For (4, 3): HCF(4, 3) = 1 (valid)
- For (5, 2): HCF(5, 2) = 1 (valid)
- For (6, 1): HCF(6, 1) = 1 (valid)

All pairs are coprime.

7. **Counting the Valid Pairs**: We have found the following valid pairs:
- (1, 6)
- (2, 5)
- (3, 4)

Each pair can be arranged in two ways (e.g., (1, 6) and (6, 1)), but since we are looking for unique pairs, we only count each pair once.

8. **Conclusion**: The total number of unique pairs \( (x, y) \) such that their sum is 7 and their HCF is 1 is 3.

### Final Answer:
The total number of such pairs is **3**.

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