In a two - digit numbers, the digit at units place is 4 more than the digit at tens place. Find the two digit number.
I. Sum of their digits is 10
II. The difference between the number and the number obtained by interchanging the position of the digits is 36.

Answer

To solve the problem step by step, we will define the two-digit number and use the given conditions to find the digits.

### Step 1: Define the digits
Let the digit at the tens place be \( x \) and the digit at the units place be \( y \). According to the problem, we have:
- The digit at the units place is 4 more than the digit at the tens place.

This gives us our first equation:
\[ y = x + 4 \]

### Step 2: Use the first condition
We are given two conditions. The first condition states that the sum of the digits is 10:
\[ x + y = 10 \]

### Step 3: Substitute the value of \( y \)
Now, we can substitute the value of \( y \) from the first equation into the second equation:
\[ x + (x + 4) = 10 \]

### Step 4: Simplify the equation
Combine like terms:
\[ 2x + 4 = 10 \]

### Step 5: Solve for \( x \)
Now, isolate \( x \):
\[ 2x = 10 - 4 \]
\[ 2x = 6 \]
\[ x = 3 \]

### Step 6: Find \( y \)
Now that we have \( x \), we can find \( y \):
\[ y = x + 4 = 3 + 4 = 7 \]

### Step 7: Form the two-digit number
The two-digit number can be formed using \( x \) and \( y \):
\[ \text{Two-digit number} = 10x + y = 10(3) + 7 = 30 + 7 = 37 \]

### Step 8: Verify the second condition
The second condition states that the difference between the number and the number obtained by interchanging the digits is 36. The number obtained by interchanging the digits is:
\[ 10y + x = 10(7) + 3 = 70 + 3 = 73 \]

Now, calculate the difference:
\[ |37 - 73| = | -36 | = 36 \]

Since both conditions are satisfied, the two-digit number is:
\[ \boxed{37} \]

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