Finding the Two-Digit Number: Three Times the Product of Digits

The problem asks us to find a two-digit number that is exactly three times the product of its digits. Let's represent a two-digit number using its digits.

Representing a Two-Digit Number

A two-digit number can be written in terms of its tens digit and units digit. If the tens digit is $a$ and the units digit is $b$, the number can be represented as $10a + b$. For a two-digit number, the tens digit $a$ must be an integer from 1 to 9, and the units digit $b$ must be an integer from 0 to 9.

Setting up the Equation for the Digit Problem

The product of the digits is $a \times b$. The problem states that the number is exactly three times the product of its digits. We can write this as an equation:

$$10a + b = 3 \times (a \times b)$$

Checking the Given Options

We are given four options. We will check each option to see if it satisfies the equation $10a + b = 3ab$.

Option 1: Checking the Number 36

• The number is 36.
• Tens digit $a = 3$.
• Units digit $b = 6$.
• The number is $10(3) + 6 = 30 + 6 = 36$.
• The product of digits is $a \times b = 3 \times 6 = 18$.
• Three times the product of digits is $3 \times 18 = 54$.
• Is $36 = 54$? No. So, 36 is not the answer.

Option 2: Checking the Number 12

• The number is 12.
• Tens digit $a = 1$.
• Units digit $b = 2$.
• The number is $10(1) + 2 = 10 + 2 = 12$.
• The product of digits is $a \times b = 1 \times 2 = 2$.
• Three times the product of digits is $3 \times 2 = 6$.
• Is $12 = 6$? No. So, 12 is not the answer.

Option 3: Checking the Number 48

• The number is 48.
• Tens digit $a = 4$.
• Units digit $b = 8$.
• The number is $10(4) + 8 = 40 + 8 = 48$.
• The product of digits is $a \times b = 4 \times 8 = 32$.
• Three times the product of digits is $3 \times 32 = 96$.
• Is $48 = 96$? No. So, 48 is not the answer.

Option 4: Checking the Number 24

• The number is 24.
• Tens digit $a = 2$.
• Units digit $b = 4$.
• The number is $10(2) + 4 = 20 + 4 = 24$.
• The product of digits is $a \times b = 2 \times 4 = 8$.
• Three times the product of digits is $3 \times 8 = 24$.
• Is $24 = 24$? Yes. So, 24 satisfies the condition.

Conclusion on Finding the Two-Digit Number

Based on checking all the given options against the condition that the two-digit number is exactly three times the product of its digits ($10a + b = 3ab$), we found that only the number 24 satisfies this condition.

Revision Table: Key Concepts for Digit Problems

Additional Information: Understanding Number and Digit Problems

Problems involving the digits of a number are common in algebra and number theory. These problems often require you to represent the number based on the place values of its digits and then set up equations based on the given conditions. For instance, if a three-digit number has digits $a, b, c$ (from hundreds to units place), the number is $100a + 10b + c$. The sum of its digits is $a+b+c$, and the product is $a \times b \times c$. Different conditions, such as reversing the digits or relationships between the number and the sum/product of its digits, lead to various types of equations.

Solving these problems typically involves:

• Assigning variables to the digits.
• Writing the number in terms of these variables and their place values.
• Formulating algebraic equations based on the problem statement.
• Solving the equations to find the values of the digits.
• Reconstructing the number using the found digits.

Sometimes, as shown in this problem, checking the given options against the derived condition is the quickest way to find the solution, especially in multiple-choice questions.