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To solve this problem, let's define the two numbers as \(x\) and \(y\). We have the following conditions:

1. The product of the two numbers is 108:
   \[
   x \cdot y = 108
   \]

2. The sum of the first number plus three times the second number should be minimized:
   \[
   S = x + 3y
   \]

### Step-by-Step Solution:

1. **Express \(x\) in terms of \(y\)** using the product condition:
   \[
   x = \frac{108}{y}
   \]

2. **Substitute into the sum expression**:
   \[
   S = \frac{108}{y} + 3y
   \]

3. To find the minimum value of \(S\), we can use calculus. Take the derivative of \(S\) with respect to \(y\) and set it to zero:
   \[
   \frac{dS}{dy} = -\frac{108}{y^2} + 3 = 0
   \]

4. **Solve for \(y\)**:
   \[
   \frac{108}{y^2} = 3
   \]
   \[
   y^2 = \frac{108}{3} = 36
   \]
   \[
   y = 6
   \]

   Since we're dealing with positive numbers (as the product is positive), we take \(y = 6\).

5. **Find \(x\)**:
   \[
   x = \frac{108}{6} = 18
   \]

6. **Verify the answer**:
   The pair \((6, 18)\) satisfies the product condition \(6 \cdot 18 = 108\). The expression \(S = 6 + 3 \cdot 18 = 60\) is the minimum for this combination.

### Final Answer:
The correct answer is **(c) 6 and 18**.

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. How do you find critical points of a function using derivatives?
2. What is the difference between a minimum and a maximum in calculus?
3. How do you verify if a critical point is a minimum or maximum?
4. What other methods can be used to solve optimization problems without calculus?
5. How would this problem change if the product condition was different?

**Tip:** When minimizing or maximizing a function, always check if you need to use derivatives, or if simpler methods like completing the square could also work.

Find two numbers that satisfy the given requirements. The product is 108 and the sum of the first number plus three times the second number is a minimum.

Learn how to solve an optimization problem where the product of two numbers is 108, and the sum of the first number plus three times the second number is minimized. This problem demonstrates key algebraic concepts and optimization using derivatives. Suitable for students in Grades 10-12.

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