In a two digit number, the unit's digit exceeds its ten's digit by 2. The product of the given number and the sum of its digits is equal to 144. Find the number.

Answer

To solve the problem step by step, we will define the variables, set up equations based on the given conditions, and then solve for the two-digit number.

### Step 1: Define the variables
Let:
- \( y \) = the ten's digit of the two-digit number
- \( x \) = the unit's digit of the two-digit number

### Step 2: Set up the first equation
According to the problem, the unit's digit exceeds the ten's digit by 2. This can be expressed as:
\[ x = y + 2 \]
This is our **Equation 1**.

### Step 3: Set up the second equation
The two-digit number can be expressed as:
\[ \text{Number} = 10y + x \]
The sum of the digits is:
\[ \text{Sum of digits} = x + y \]

According to the problem, the product of the number and the sum of its digits is equal to 144:
\[ (10y + x)(x + y) = 144 \]
Substituting \( x \) from Equation 1 into this equation gives:
\[ (10y + (y + 2))((y + 2) + y) = 144 \]
This simplifies to:
\[ (10y + y + 2)(y + 2 + y) = 144 \]
\[ (11y + 2)(2y + 2) = 144 \]

### Step 4: Simplify the equation
Now we can simplify further:
\[ (11y + 2)(2y + 2) = 144 \]
Expanding this:
\[ 22y^2 + 22y + 4y + 4 = 144 \]
Combining like terms:
\[ 22y^2 + 26y + 4 = 144 \]
Subtracting 144 from both sides:
\[ 22y^2 + 26y + 4 - 144 = 0 \]
\[ 22y^2 + 26y - 140 = 0 \]

### Step 5: Simplify the quadratic equation
To simplify, we can divide the entire equation by 2:
\[ 11y^2 + 13y - 70 = 0 \]

### Step 6: Factor the quadratic equation
Now we need to factor the quadratic equation:
We are looking for two numbers that multiply to \( 11 \times (-70) = -770 \) and add to \( 13 \). The numbers are \( 35 \) and \( -22 \). Thus, we can write:
\[ 11y^2 + 35y - 22y - 70 = 0 \]
Grouping the terms:
\[ (11y^2 + 35y) + (-22y - 70) = 0 \]
Factoring by grouping:
\[ 11y(y + 3) - 22(y + 3) = 0 \]
Factoring out \( (y + 3) \):
\[ (y + 3)(11y - 22) = 0 \]

### Step 7: Solve for \( y \)
Setting each factor to zero gives:
1. \( y + 3 = 0 \) → \( y = -3 \) (not valid since \( y \) must be a digit)
2. \( 11y - 22 = 0 \) → \( 11y = 22 \) → \( y = 2 \)

### Step 8: Find \( x \)
Now we can find \( x \) using Equation 1:
\[ x = y + 2 = 2 + 2 = 4 \]

### Step 9: Form the two-digit number
The two-digit number is:
\[ \text{Number} = 10y + x = 10(2) + 4 = 20 + 4 = 24 \]

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

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