The LCM of two numbers is 495 and their HCF is 5. If the sum of the numbers is 100, then their difference is :

Options

A. 10
B. 46
C. 70
D. 90

Answer

To solve the problem step by step, we need to find the difference between two numbers given their LCM, HCF, and the sum of the numbers.

### Step 1: Understand the relationship between LCM, HCF, and the product of two numbers.
The product of two numbers can be expressed as:
\[ \text{Product of two numbers} = \text{LCM} \times \text{HCF} \]

### Step 2: Substitute the known values.
Given:
- LCM = 495
- HCF = 5

Using the formula:
\[ \text{Product of two numbers} = 495 \times 5 \]

### Step 3: Calculate the product.
Now, calculate the product:
\[ 495 \times 5 = 2475 \]

### Step 4: Let the two numbers be \( x \) and \( y \).
We know:
- \( x + y = 100 \) (the sum of the numbers)
- \( x \times y = 2475 \) (the product of the numbers)

### Step 5: Use the equations to form a quadratic equation.
From the sum and product, we can form the quadratic equation:
\[ t^2 - (x+y)t + xy = 0 \]
Substituting the values:
\[ t^2 - 100t + 2475 = 0 \]

### Step 6: Solve the quadratic equation.
To solve for \( t \), we can use the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1, b = -100, c = 2475 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-100)^2 - 4 \times 1 \times 2475 \]
\[ = 10000 - 9900 = 100 \]

Now, substituting into the quadratic formula:
\[ t = \frac{100 \pm \sqrt{100}}{2} \]
\[ = \frac{100 \pm 10}{2} \]

Calculating the two possible values for \( t \):
1. \( t = \frac{110}{2} = 55 \)
2. \( t = \frac{90}{2} = 45 \)

### Step 7: Identify the two numbers.
Thus, the two numbers are:
- \( x = 55 \)
- \( y = 45 \)

### Step 8: Calculate the difference.
Now, we find the difference:
\[ x - y = 55 - 45 = 10 \]

### Final Answer:
The difference between the two numbers is **10**.

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