A two digit number is obtained by multiplying the sum of the digits by 8. Also, it is obtained by multiplying the difference of the digits by 14 and adding 2. Find the number.

Answer

To solve the problem, we need to find a two-digit number based on the conditions provided. Let's break down the steps systematically.

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

Thus, the two-digit number can be expressed as:
\[ 10x + y \]

### Step 2: Set Up the First Equation
According to the problem, the two-digit number is obtained by multiplying the sum of the digits by 8. Therefore, we can write:
\[ 10x + y = 8(x + y) \]
Expanding this gives:
\[ 10x + y = 8x + 8y \]
Rearranging the equation leads to:
\[ 10x + y - 8x - 8y = 0 \implies 2x - 7y = 0 \]
This is our **first equation**:
\[ 2x - 7y = 0 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that the two-digit number is obtained by multiplying the difference of the digits by 14 and adding 2. Thus, we can write:
\[ 10x + y = 14(x - y) + 2 \]
Expanding this gives:
\[ 10x + y = 14x - 14y + 2 \]
Rearranging the equation leads to:
\[ 10x + y - 14x + 14y - 2 = 0 \implies -4x + 15y - 2 = 0 \]
This is our **second equation**:
\[ 4x - 15y = -2 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( 2x - 7y = 0 \)
2. \( 4x - 15y = -2 \)

From Equation 1, we can express \( x \) in terms of \( y \):
\[ 2x = 7y \implies x = \frac{7y}{2} \]

### Step 5: Substitute in the Second Equation
Substituting \( x = \frac{7y}{2} \) into Equation 2:
\[ 4\left(\frac{7y}{2}\right) - 15y = -2 \]
This simplifies to:
\[ 14y - 15y = -2 \implies -y = -2 \implies y = 2 \]

### Step 6: Find the Value of \( x \)
Now that we have \( y = 2 \), we can find \( x \):
\[ x = \frac{7(2)}{2} = 7 \]

### Step 7: Form the Two-Digit Number
Now we can find the two-digit number:
\[ 10x + y = 10(7) + 2 = 70 + 2 = 72 \]

### Final Answer
Thus, the two-digit number is:
\[ \boxed{72} \]