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

Options

A. 115
B. 122
C. 124
D. 138

Answer

To solve the problem step by step, we will use the information provided about the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the two numbers.

### Step 1: Define Variables
Let the HCF of the two numbers be \( x \). According to the problem, the LCM is 12 times the HCF. Therefore, we can express the LCM as:
\[ \text{LCM} = 12x \]

### Step 2: Set Up the Equation for the Sum
The problem states that the sum of the HCF and LCM is 403. We can write this as:
\[ x + 12x = 403 \]

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

### Step 4: Solve for HCF
Now, divide both sides by 13 to find \( x \):
\[ x = \frac{403}{13} \]
Calculating this gives:
\[ x = 31 \]
So, the HCF is 31.

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

### Step 6: Use the Relationship Between HCF, LCM, and the Numbers
We know that for any two numbers \( a \) and \( b \):
\[ \text{HCF} \times \text{LCM} = a \times b \]
Given one number \( a = 93 \), we can substitute the values we have:
\[ 31 \times 372 = 93 \times b \]

### Step 7: Solve for the Other Number
Now, calculate \( 31 \times 372 \):
\[ 31 \times 372 = 11532 \]
Now, set up the equation:
\[ 11532 = 93 \times b \]
To find \( b \), divide both sides by 93:
\[ b = \frac{11532}{93} \]
Calculating this gives:
\[ b = 124 \]

### Conclusion
The other number is \( 124 \).

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