Sum of two numbers is 384. HCF of the numbers is 48. The difference of the number is:

Options

A. 100
B. 192
C. 288
D. 336

Answer

To solve the problem step by step, we will use the information given about the sum and the HCF of the two numbers.

### Step 1: Understand the problem
We know:
- The sum of two numbers is 384.
- The HCF of the two numbers is 48.

Let the two numbers be \( a \) and \( b \).

### Step 2: Express the numbers in terms of HCF
Since the HCF of the two numbers is 48, we can express the two numbers as:
- \( a = 48x \)
- \( b = 48y \)

where \( x \) and \( y \) are coprime integers (they have no common factors other than 1).

### Step 3: Set up the equation for the sum
From the problem, we know:
\[ a + b = 384 \]
Substituting the expressions for \( a \) and \( b \):
\[ 48x + 48y = 384 \]

### Step 4: Simplify the equation
We can factor out 48 from the left side:
\[ 48(x + y) = 384 \]
Now, divide both sides by 48:
\[ x + y = \frac{384}{48} \]
Calculating the right side:
\[ x + y = 8 \]

### Step 5: Find pairs of coprime integers
Now we need to find pairs of integers \( (x, y) \) such that \( x + y = 8 \) and \( x \) and \( y \) are coprime. The possible pairs are:
1. \( (1, 7) \)
2. \( (3, 5) \)
3. \( (5, 3) \)
4. \( (7, 1) \)

### Step 6: Calculate the difference for each pair
Now we will calculate the difference \( |a - b| \) for each pair:
- For \( (1, 7) \):
\[ a = 48 \times 1 = 48, \quad b = 48 \times 7 = 336 \]
\[ |a - b| = |48 - 336| = 288 \]

- For \( (3, 5) \):
\[ a = 48 \times 3 = 144, \quad b = 48 \times 5 = 240 \]
\[ |a - b| = |144 - 240| = 96 \]

- For \( (5, 3) \):
\[ a = 48 \times 5 = 240, \quad b = 48 \times 3 = 144 \]
\[ |a - b| = |240 - 144| = 96 \]

- For \( (7, 1) \):
\[ a = 48 \times 7 = 336, \quad b = 48 \times 1 = 48 \]
\[ |a - b| = |336 - 48| = 288 \]

### Step 7: Conclusion
The possible differences are 288 and 96. Since the question asks for the difference of the numbers, we see that the valid differences are 288 and 96, but since the options provided include 288, we conclude that:

**The difference of the two numbers is 288.**