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. 5
B. 25
C. 100
D. 125

Answer

To solve the problem step by step, we will use the relationship between the LCM (Least Common Multiple) and HCF (Highest Common Factor) of two numbers.

### Step 1: Define the variables
Let:
- HCF = \( h \)
- LCM = \( l \)

According to the problem:
1. \( l = 4h \) (The LCM is 4 times the HCF)
2. \( l + h = 125 \) (The sum of LCM and HCF is 125)

### Step 2: Substitute the LCM in the sum equation
From the first equation, we can substitute \( l \) in the second equation:
\[ 4h + h = 125 \]

### Step 3: Combine like terms
Combine the terms on the left side:
\[ 5h = 125 \]

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

### Step 5: Find the LCM
Now that we have the HCF, we can find the LCM using the first equation:
\[ l = 4h = 4 \times 25 = 100 \]

### Step 6: Use the relationship between LCM, HCF, and the numbers
We know that for two numbers \( N_1 \) and \( N_2 \):
\[ N_1 \times N_2 = LCM \times HCF \]
Given \( N_1 = 100 \), we can substitute the values:
\[ 100 \times N_2 = 100 \times 25 \]

### Step 7: Solve for the other number
Now, divide both sides by 100:
\[ N_2 = 25 \]

Thus, the other number is **25**.