The product of digits of a 2-digit number is 24. If we add 45 to the number, the new number obtained is a number formed by interchanging the digits. What is the original number?

Options

A. a) 54
B. b) 83
C. c) 38
D. d) 45

Answer

To solve the problem step by step, we will denote the two-digit number as \( xy \), where \( x \) is the tens digit and \( y \) is the units digit.

### Step 1: Set up the equations
1. The product of the digits is given as:
\[ x \cdot y = 24 \]
2. When we add 45 to the number \( 10x + y \), we obtain a new number which is formed by interchanging the digits, represented as \( 10y + x \). Therefore, we can write the equation:
\[ 10x + y + 45 = 10y + x \]

### Step 2: Rearranging the second equation
Let's rearrange the second equation:
\[ 10x + y + 45 = 10y + x \]
Subtract \( x \) and \( y \) from both sides:
\[ 10x - x + y - y + 45 = 10y - y \]
This simplifies to:
\[ 9x + 45 = 9y \]
Now, divide the entire equation by 9:
\[ x + 5 = y \]
Thus, we have:
\[ y - x = 5 \quad \text{(Equation 1)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( xy = 24 \) (Equation 2)
2. \( y - x = 5 \) (Equation 1)

From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = x + 5 \]

### Step 4: Substitute into the product equation
Substituting \( y \) into Equation 2:
\[ x(x + 5) = 24 \]
Expanding this gives:
\[ x^2 + 5x - 24 = 0 \]

### Step 5: Factor the quadratic equation
Now, we need to factor the quadratic equation:
\[ x^2 + 5x - 24 = 0 \]
We look for two numbers that multiply to \(-24\) and add to \(5\). The numbers \(8\) and \(-3\) satisfy this:
\[ (x + 8)(x - 3) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives:
\[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \quad \text{(not valid since } x \text{ must be a digit)} \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

### Step 7: Find \( y \)
Now substitute \( x = 3 \) back into the equation for \( y \):
\[ y = x + 5 = 3 + 5 = 8 \]

### Step 8: Write the original number
Thus, the original two-digit number is:
\[ 10x + y = 10(3) + 8 = 30 + 8 = 38 \]

### Conclusion
The original number is **38**.