Let's go through this step-by-step.

Given:

1. The LCM of two numbers is 12 times their HCF.
2. The sum of HCF and LCM is 403.
3. The first number is 93.

We need to find the second number.

Let:

• $H$ be the HCF of the two numbers.
• $L$ be the LCM of the two numbers.

From the problem, we have the following relationships:

1. $L = 12 \times H$
2. $H + L = 403$

Step 1: Substitute $L = 12H$ into $H + L = 403$

$H + 12H = 403$ $13H = 403$ $H = \frac{403}{13} = 31$

So, $H = 31$.

Step 2: Find $L$ using $L = 12H$

$L = 12 \times 31 = 372$

Now we know:

• HCF ($H$) = 31
• LCM ($L$) = 372

Step 3: Use the relationship between HCF, LCM, and the two numbers

For two numbers $a$ and $b$, the product of the numbers is given by: $a \times b = \text{HCF} \times \text{LCM}$ Substituting the values we have: $93 \times b = 31 \times 372$

Calculating $31 \times 372$: $31 \times 372 = 11532$

Thus: $93 \times b = 11532$ $b = \frac{11532}{93} = 124$

Conclusion:

The second number is 124.

Would you like more details on any of these steps, or have any further questions?

Here are some related questions for further exploration:

1. What is the relationship between HCF and LCM of two numbers in general?
2. How can the formula $\text{HCF} \times \text{LCM} = a \times b$ be derived?
3. If two numbers have their LCM equal to their HCF, what does it imply about the numbers?
4. How would the solution change if we knew only the LCM or HCF alone?
5. Why does knowing one number's value and the LCM-HCF relationship help in finding the other number?

Tip: When given HCF and LCM relationships in problems, try using substitution techniques to simplify and solve the equations directly.