The sum of the digits of a 2-digits number is 6. The number obtained by interchangind its digits is 18 more than the original number. Find the original number.

Answer

To solve the problem step by step, let's denote the two-digit number as \(10X + Y\), where \(X\) is the digit in the tens place and \(Y\) is the digit in the units place.

### Step 1: Set up the equations based on the problem statement.
1. The sum of the digits is 6:
\[ X + Y = 6 \quad \text{(Equation 1)} \]

2. The number obtained by interchanging the digits is 18 more than the original number:
\[ 10Y + X = (10X + Y) + 18 \quad \text{(Equation 2)} \]

### Step 2: Simplify Equation 2.
Rearranging Equation 2:
\[ 10Y + X - 10X - Y = 18 \]
This simplifies to:
\[ 9Y - 9X = 18 \]
Dividing through by 9 gives:
\[ Y - X = 2 \quad \text{(Equation 3)} \]

### Step 3: Solve the system of equations.
Now we have two equations:
1. \(X + Y = 6\) (Equation 1)
2. \(Y - X = 2\) (Equation 3)

We can solve these equations simultaneously.

From Equation 3, we can express \(Y\) in terms of \(X\):
\[ Y = X + 2 \]

### Step 4: Substitute \(Y\) in Equation 1.
Substituting \(Y\) in Equation 1:
\[ X + (X + 2) = 6 \]
This simplifies to:
\[ 2X + 2 = 6 \]
Subtracting 2 from both sides:
\[ 2X = 4 \]
Dividing by 2:
\[ X = 2 \]

### Step 5: Find \(Y\).
Now substitute \(X\) back into the expression for \(Y\):
\[ Y = 2 + 2 = 4 \]

### Step 6: Write the original number.
The original two-digit number is:
\[ 10X + Y = 10(2) + 4 = 20 + 4 = 24 \]

### Step 7: Verify the solution.
1. The sum of the digits \(2 + 4 = 6\) (correct).
2. The number obtained by interchanging the digits is \(42\), and \(42 - 24 = 18\) (correct).

Thus, the original number is **24**.