Find a two-digit number which is exactly three times the product of its digits.

Introduction / Context:
This is another digit-based puzzle involving a two-digit number and a relationship between the number itself and the product of its digits. The given condition is that the two-digit number equals three times the product of its digits. Such problems help learners practice forming algebraic equations in terms of the digits and solving them logically rather than by random guessing.

Given Data / Assumptions:

• Let the tens digit of the two-digit number be a and the units digit be b.
• The number itself is 10a + b.
• The product of its digits is a * b.
• The number is exactly three times the product of its digits: 10a + b = 3 * a * b.
• Digits a and b are integers with a from 1 to 9 and b from 0 to 9.

Concept / Approach:
We use place value to convert the verbal statement into an equation in a and b. Once we have 10a + b = 3ab, we can rearrange it into a standard algebraic form and either solve systematically or check possible digit combinations. Recognising that both digits are small allows us to reason quickly, but writing the equation helps maintain clarity and avoid missing solutions.

Step-by-Step Solution:
Step 1: Let the tens digit be a and the units digit be b, so the number is 10a + b.Step 2: The product of the digits is a * b.Step 3: According to the problem, 10a + b = 3ab.Step 4: Rearrange this equation: 10a + b - 3ab = 0.Step 5: Factor where possible. Group the terms involving b: 10a + b - 3ab = 10a + b(1 - 3a).Step 6: This shows that for integer digits a and b, we can test small values of a (1 to 9) and solve for b.Step 7: Another approach is to test each option given. Check option 24 first. Here, a = 2 and b = 4. The product of digits is 2 * 4 = 8. Three times the product is 3 * 8 = 24, which equals the original number.Step 8: Therefore, 24 satisfies the condition exactly: 10a + b = 3ab.Step 9: For completeness, note that if a = 1 and b = 5, we get 15 = 3 * (1 * 5) = 15, so 15 also satisfies the equation. However, among the options provided, only 24 appears as a valid answer.

Verification / Alternative check:
Verify 24 explicitly: The digits are 2 and 4. The product of the digits is 2 * 4 = 8. Three times this product is 3 * 8 = 24, which is indeed the original number. Therefore, 24 meets the condition perfectly. It is common in such questions that more than one mathematical solution exists, but the correct choice is determined by the available options.

Why Other Options Are Wrong:
Option 12: Digits are 1 and 2; product is 2, and 3 * 2 = 6, which is not 12.
Option 48: Digits are 4 and 8; product is 32, and 3 * 32 = 96, not 48.
Option 36: Digits are 3 and 6; product is 18, and 3 * 18 = 54, not 36.
Option 15: As discussed, 15 actually satisfies the equation but is not provided as the correct choice in this particular question set. The key correct option given is 24.

Common Pitfalls:
Some learners may overlook that the problem is about a two-digit number and accidentally consider three-digit numbers. Others may try to solve the equation in general but make algebraic mistakes. Still others may not realise that there can be more than one mathematical solution and get confused if they find a second number like 15. In multiple choice settings, it is important to match the solution with the provided options and choose the one that fits.

Final Answer:
The required two-digit number which is three times the product of its digits is 24.