A two digit number is obtained by either multiplying the sum of the digits by 8 and subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.

Answer

To solve the problem, we need to find a two-digit number based on the conditions given. Let's denote the digits of the two-digit number as follows:

- Let \( Y \) be the ten's digit.
- Let \( X \) be the one's digit.

The two-digit number can be expressed as:
\[ 10Y + X \]

### Step 1: Set up the equations based on the problem statement

According to the problem, the two-digit number can be obtained in two ways:

1. By multiplying the sum of the digits by 8 and subtracting 5:
\[ 10Y + X = 8(X + Y) - 5 \]

2. By multiplying the difference of the digits by 16 and adding 3:
\[ 10Y + X = 16(Y - X) + 3 \]

### Step 2: Simplify the first equation

Starting with the first equation:
\[ 10Y + X = 8(X + Y) - 5 \]
Expanding the right side:
\[ 10Y + X = 8X + 8Y - 5 \]
Rearranging gives:
\[ 10Y + X - 8X - 8Y = -5 \]
Combining like terms:
\[ 2Y - 7X = -5 \quad \text{(Equation 1)} \]

### Step 3: Simplify the second equation

Now, simplifying the second equation:
\[ 10Y + X = 16(Y - X) + 3 \]
Expanding the right side:
\[ 10Y + X = 16Y - 16X + 3 \]
Rearranging gives:
\[ 10Y + X - 16Y + 16X = 3 \]
Combining like terms:
\[ -6Y + 17X = 3 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations

Now we have a system of equations:
1. \( 2Y - 7X = -5 \)
2. \( -6Y + 17X = 3 \)

To eliminate \( Y \), we can multiply Equation 1 by 3:
\[ 6Y - 21X = -15 \quad \text{(Equation 3)} \]

Now, we can add Equation 2 and Equation 3:
\[ (-6Y + 17X) + (6Y - 21X) = 3 - 15 \]
This simplifies to:
\[ -4X = -12 \]
Dividing by -4 gives:
\[ X = 3 \]

### Step 5: Substitute \( X \) back to find \( Y \)

Now we substitute \( X = 3 \) back into Equation 1:
\[ 2Y - 7(3) = -5 \]
This simplifies to:
\[ 2Y - 21 = -5 \]
Adding 21 to both sides gives:
\[ 2Y = 16 \]
Dividing by 2 gives:
\[ Y = 8 \]

### Step 6: Form the two-digit number

Now we have:
- \( Y = 8 \) (ten's digit)
- \( X = 3 \) (one's digit)

Thus, the two-digit number is:
\[ 10Y + X = 10(8) + 3 = 80 + 3 = 83 \]

### Final Answer
The two-digit number is **83**.