The LCM of two numbers is 12 times their HCF. The sum of HCF and LCM is 403. If one of the numbers is 93, then the other number is

Options

A. 124
B. 128
C. 134
D. 138

Answer

To solve the problem step by step, we will use the information given in the question.

### Step 1: Define the variables
Let the HCF of the two numbers be \( x \). According to the problem, the LCM of the two numbers is \( 12 \times x \).

### Step 2: Set up the equation for the sum of HCF and LCM
We know that the sum of HCF and LCM is given as 403. Therefore, we can write the equation:
\[ x + 12x = 403 \]

### Step 3: Simplify the equation
Combine like terms:
\[ 13x = 403 \]

### Step 4: Solve for HCF
Now, we can solve for \( x \) by dividing both sides by 13:
\[ x = \frac{403}{13} = 31 \]
So, the HCF is \( 31 \).

### Step 5: Calculate the LCM
Now that we have the HCF, we can calculate the LCM:
\[ \text{LCM} = 12 \times HCF = 12 \times 31 = 372 \]

### 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:
\[ \text{Product of the two numbers} = HCF \times LCM = 31 \times 372 \]

### Step 7: Calculate the product
Calculating the product:
\[ 31 \times 372 = 11532 \]

### Step 8: Use the known number to find the other number
Let one of the numbers be \( 93 \) (as given in the question). Let the other number be \( y \). Therefore, we have:
\[ 93 \times y = 11532 \]

### Step 9: Solve for \( y \)
Now, we can solve for \( y \) by dividing both sides by 93:
\[ y = \frac{11532}{93} = 124 \]

### Conclusion
Thus, the other number is \( 124 \).