The LCM is 15 times of HCF in two numbers. If the sum of LCM and HCF is 480 and one numbers is 90, find the 2nd number.

Options

A. 100
B. 160
C. 150
D. 200

Answer

To solve the problem step by step, we will use the relationship between LCM (Least Common Multiple) and HCF (Highest Common Factor) along with the given information.

### Step 1: Define the variables
Let:
- HCF = x
- LCM = 15x (since the LCM is 15 times the HCF)

### Step 2: Set up the equation for the sum of LCM and HCF
According to the problem, the sum of LCM and HCF is 480:
\[ LCM + HCF = 480 \]
Substituting the values we defined:
\[ 15x + x = 480 \]

### Step 3: Simplify the equation
Combine like terms:
\[ 16x = 480 \]

### Step 4: Solve for HCF (x)
Now, divide both sides by 16 to find x:
\[ x = \frac{480}{16} = 30 \]
So, HCF = 30.

### Step 5: Find LCM
Now, substitute x back to find LCM:
\[ LCM = 15x = 15 \times 30 = 450 \]

### Step 6: Use the relationship between LCM, HCF, and the two numbers
We know that for two numbers \( a \) and \( b \):
\[ LCM \times HCF = a \times b \]
Given that one number (let's say \( a \)) is 90, we can substitute the values:
\[ 450 \times 30 = 90 \times b \]

### Step 7: Solve for the second number (b)
Now, we can solve for \( b \):
\[ 13500 = 90 \times b \]
Dividing both sides by 90:
\[ b = \frac{13500}{90} = 150 \]

### Conclusion
The second number is **150**.

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