The sum of the two digits of a two-digit number is 15 and the difference between the two digits of the two digit number is 3. What is the product of the two digits of the two digit number ?

Options

A. 72
B. 56
C. 54
D. Cannot be determined

Explanation

To solve the problem step by step, we will define the two digits of the two-digit number and set up equations based on the information provided.

### Step 1: Define the Variables
Let the two digits of the two-digit number be \( x \) (the tens digit) and \( y \) (the units digit).

### Step 2: Set Up the Equations
According to the problem, we have two conditions:
1. The sum of the two digits is 15:
\[ x + y = 15 \quad \text{(Equation 1)} \]
2. The difference between the two digits is 3:
\[ x - y = 3 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
To solve these equations, we can add Equation 1 and Equation 2 together:
\[ (x + y) + (x - y) = 15 + 3 \]
This simplifies to:
\[ 2x = 18 \]
Now, divide both sides by 2:
\[ x = 9 \]

### Step 4: Find the Value of \( y \)
Now that we have \( x \), we can substitute it back into Equation 1 to find \( y \):
\[ 9 + y = 15 \]
Subtract 9 from both sides:
\[ y = 15 - 9 = 6 \]

### Step 5: Calculate the Product of the Digits
Now that we have both digits, \( x = 9 \) and \( y = 6 \), we can find the product:
\[ \text{Product} = x \cdot y = 9 \cdot 6 = 54 \]

### Final Answer
The product of the two digits of the two-digit number is \( 54 \).

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