The LCM of two numbers is 4 times their HCF. The sum of LCM and HCF is 125. If one of the number is 100, then the other number is?

Options

A. a. 5
B. b. 25
C. c.100
D. d. 125

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define the variables
Let the HCF (Highest Common Factor) be \( x \). According to the problem, the LCM (Lowest Common Multiple) is 4 times the HCF. Therefore, we can express the LCM as:
\[ \text{LCM} = 4x \]

### Step 2: Set up the equation based on the sum of LCM and HCF
We know that the sum of the LCM and HCF is 125. Therefore, we can write the equation:
\[ \text{LCM} + \text{HCF} = 125 \]
Substituting the values we have:
\[ 4x + x = 125 \]

### Step 3: Simplify the equation
Combine like terms:
\[ 5x = 125 \]

### Step 4: Solve for HCF
Now, divide both sides by 5 to find \( x \):
\[ x = \frac{125}{5} = 25 \]
So, the HCF is 25.

### Step 5: Calculate the LCM
Now, substitute \( x \) back into the equation for LCM:
\[ \text{LCM} = 4x = 4 \times 25 = 100 \]
So, the LCM is 100.

### Step 6: Use the relationship between HCF, LCM, and the two numbers
We know that the product of the two numbers is equal to the product of their HCF and LCM:
\[ \text{Number 1} \times \text{Number 2} = \text{HCF} \times \text{LCM} \]
Let’s denote the two numbers as \( A \) and \( B \). Given that one of the numbers \( A = 100 \) and the HCF is 25, the LCM is 100. We can set up the equation:
\[ 100 \times B = 25 \times 100 \]

### Step 7: Solve for the other number
Now, simplify the equation:
\[ 100B = 2500 \]
Divide both sides by 100:
\[ B = \frac{2500}{100} = 25 \]

### Conclusion
Thus, the other number is:
\[ \boxed{25} \]