The sum of two numbers is 684 and their HCF is 57. The number of possible pairs of such numbers is:

Options

A. 2
B. 3
C. 4
D. None of these

Answer

To solve the problem step by step, we need to determine the number of possible pairs of numbers whose sum is 684 and whose highest common factor (HCF) is 57.

### Step 1: Set up the equations
Let the two numbers be \( x \) and \( y \). According to the problem:
\[ x + y = 684 \quad \text{(Equation 1)} \]
Also, since their HCF is 57, we can express \( x \) and \( y \) in terms of their multiples of 57:
\[ x = 57a \quad \text{and} \quad y = 57b \]
where \( a \) and \( b \) are integers.

### Step 2: Substitute into the equation
Substituting \( x \) and \( y \) into Equation 1 gives:
\[ 57a + 57b = 684 \]
Factoring out 57, we have:
\[ 57(a + b) = 684 \]

### Step 3: Simplify the equation
To isolate \( a + b \), divide both sides by 57:
\[ a + b = \frac{684}{57} \]
Calculating the right side:
\[ a + b = 12 \]

### Step 4: Determine pairs of \( (a, b) \)
Now, we need to find pairs of positive integers \( (a, b) \) such that:
\[ a + b = 12 \]
The pairs of integers that satisfy this equation are:
1. \( (1, 11) \)
2. \( (2, 10) \)
3. \( (3, 9) \)
4. \( (4, 8) \)
5. \( (5, 7) \)
6. \( (6, 6) \)

### Step 5: Check for coprimality
Since \( x \) and \( y \) must be coprime (as their HCF is 57), we need to check which pairs \( (a, b) \) are coprime:
- \( (1, 11) \) - coprime
- \( (2, 10) \) - not coprime
- \( (3, 9) \) - not coprime
- \( (4, 8) \) - not coprime
- \( (5, 7) \) - coprime
- \( (6, 6) \) - not coprime

The coprime pairs are \( (1, 11) \) and \( (5, 7) \).

### Step 6: Calculate the actual numbers
Now we can find the actual pairs of numbers \( (x, y) \):
1. For \( (1, 11) \):
\[ x = 57 \times 1 = 57, \quad y = 57 \times 11 = 627 \]
Pair: \( (57, 627) \)

2. For \( (5, 7) \):
\[ x = 57 \times 5 = 285, \quad y = 57 \times 7 = 399 \]
Pair: \( (285, 399) \)

### Conclusion
Thus, the number of possible pairs of such numbers is **2**.