In a two digit number if it is known that its units digit exceeds its tens digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is

Options

A. 46
B. 42
C. 26
D. 24

Answer

To solve the problem step by step, we will follow the information given in the question and use algebraic expressions.

### Step 1: Define the digits of the two-digit number
Let the tens digit be \( x \). According to the problem, the units digit exceeds the tens digit by 2. Therefore, we can express the units digit as:
\[ \text{Units digit} = x + 2 \]

### Step 2: Write the two-digit number
The two-digit number can be represented as:
\[ \text{Number} = 10 \times (\text{Tens digit}) + (\text{Units digit}) = 10x + (x + 2) \]
This simplifies to:
\[ \text{Number} = 11x + 2 \]

### Step 3: Find the sum of the digits
The sum of the digits of the number is:
\[ \text{Sum of digits} = \text{Tens digit} + \text{Units digit} = x + (x + 2) = 2x + 2 \]

### Step 4: Set up the equation based on the problem statement
According to the problem, the product of the number and the sum of its digits is equal to 144. Therefore, we can write the equation:
\[ (11x + 2)(2x + 2) = 144 \]

### Step 5: Expand the equation
Now we will expand the left-hand side:
\[ (11x + 2)(2x + 2) = 22x^2 + 22x + 4 \]
Setting up the equation:
\[ 22x^2 + 22x + 4 = 144 \]

### Step 6: Rearrange the equation
Now, we will move 144 to the left side:
\[ 22x^2 + 22x + 4 - 144 = 0 \]
This simplifies to:
\[ 22x^2 + 22x - 140 = 0 \]

### Step 7: Simplify the equation
We can divide the entire equation by 2 to simplify:
\[ 11x^2 + 11x - 70 = 0 \]

### Step 8: Factor the quadratic equation
Now we will factor the quadratic equation. We need two numbers that multiply to \(-770\) (11 * -70) and add to \(11\). The factors are:
\[ (11x + 35)(x - 2) = 0 \]

### Step 9: Solve for \( x \)
Setting each factor to zero gives us:
1. \( 11x + 35 = 0 \) → \( x = -\frac{35}{11} \) (not a valid digit)
2. \( x - 2 = 0 \) → \( x = 2 \)

### Step 10: Find the units digit and the number
Since \( x = 2 \) (the tens digit), we can find the units digit:
\[ \text{Units digit} = x + 2 = 2 + 2 = 4 \]
Thus, the two-digit number is:
\[ \text{Number} = 10 \times 2 + 4 = 24 \]

### Final Answer
The two-digit number is **24**.

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