The sum of two numbers is 684 and their H.C.F is 57. The number of possible pairs of such numbers is

Options

A. 2
B. 3
C. 4
D. none

Answer

To solve the problem step by step, we need to find the number of possible pairs of two numbers whose sum is 684 and whose highest common factor (H.C.F) is 57.

### Step 1: Set up the equations
Let the two numbers be represented as:
- First number = 57x
- Second number = 57y

According to the problem, the sum of these two numbers is:
\[ 57x + 57y = 684 \]

### Step 2: Simplify the equation
We can factor out 57 from the left side:
\[ 57(x + y) = 684 \]

Now, divide both sides by 57:
\[ x + y = \frac{684}{57} \]

### Step 3: Calculate the value of \(x + y\)
Now, we perform the division:
\[ x + y = 12 \]

### Step 4: Find pairs of (x, y)
We need to find pairs of positive integers (x, y) such that:
\[ x + y = 12 \]

We can express y in terms of x:
\[ y = 12 - x \]

### Step 5: Check the H.C.F condition
The H.C.F of the two numbers \(57x\) and \(57y\) is 57. For this to hold, x and y must be coprime (i.e., their H.C.F must be 1).

Now we will check pairs of (x, y):
1. If \(x = 1\), then \(y = 11\) (H.C.F(1, 11) = 1) → Valid
2. If \(x = 2\), then \(y = 10\) (H.C.F(2, 10) = 2) → Not valid
3. If \(x = 3\), then \(y = 9\) (H.C.F(3, 9) = 3) → Not valid
4. If \(x = 4\), then \(y = 8\) (H.C.F(4, 8) = 4) → Not valid
5. If \(x = 5\), then \(y = 7\) (H.C.F(5, 7) = 1) → Valid
6. If \(x = 6\), then \(y = 6\) (H.C.F(6, 6) = 6) → Not valid

### Step 6: Count the valid pairs
From the above checks, the valid pairs are:
1. (1, 11)
2. (5, 7)

Thus, we have found **2 valid pairs** of (x, y).

### Final Answer
The number of possible pairs of such numbers is **2**.

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