A two digit number is four times the sum and three times the product of its digits.

Answer

To solve the problem step by step, we will define the two-digit number based on its digits and then use the given conditions to form equations.

### Step 1: Define the digits of the two-digit number
Let the unit digit of the number be \( x \) and the tens digit be \( y \). Therefore, the two-digit number can be expressed as:
\[ 10y + x \]

### Step 2: Set up the first equation based on the sum of the digits
According to the problem, the two-digit number is four times the sum of its digits. The sum of the digits is \( x + y \). Thus, we can write the equation:
\[ 10y + x = 4(x + y) \]

### Step 3: Simplify the first equation
Expanding the right side gives:
\[ 10y + x = 4x + 4y \]
Rearranging this equation, we get:
\[ 10y - 4y = 4x - x \]
\[ 6y = 3x \]
Dividing both sides by 3 gives:
\[ 2y = x \quad \text{(Equation 1)} \]

### Step 4: Set up the second equation based on the product of the digits
The problem also states that the two-digit number is three times the product of its digits. The product of the digits is \( xy \). Thus, we can write the second equation:
\[ 10y + x = 3xy \]

### Step 5: Substitute \( x \) from Equation 1 into the second equation
From Equation 1, we know \( x = 2y \). Substituting this into the second equation gives:
\[ 10y + 2y = 3(2y)y \]
This simplifies to:
\[ 12y = 6y^2 \]

### Step 6: Rearrange and simplify the equation
Rearranging the equation gives:
\[ 6y^2 - 12y = 0 \]
Factoring out \( 6y \) gives:
\[ 6y(y - 2) = 0 \]

### Step 7: Solve for \( y \)
Setting each factor to zero gives:
\[ 6y = 0 \quad \text{or} \quad y - 2 = 0 \]
Thus, \( y = 0 \) or \( y = 2 \). Since \( y \) represents the tens digit of a two-digit number, it cannot be 0. Therefore:
\[ y = 2 \]

### Step 8: Find \( x \) using Equation 1
Now, substituting \( y = 2 \) back into Equation 1:
\[ x = 2y = 2 \cdot 2 = 4 \]

### Step 9: Form the two-digit number
Now that we have both digits, \( x = 4 \) and \( y = 2 \), we can find the two-digit number:
\[ 10y + x = 10 \cdot 2 + 4 = 20 + 4 = 24 \]

### Final Answer
The two-digit number is:
\[ \boxed{24} \]