Let's solve the problem step by step:

Let the original two-digit number be represented as:

$10a + b$

where $a$ is the tens digit and $b$ is the ones digit.

Step 1: First condition - The sum of the digits is 8

The sum of the digits of the number is given as:

a + b = 8 \tag{1}

Step 2: Second condition - The number obtained by interchanging the digits exceeds the original number by 18

If the digits are interchanged, the new number becomes:

$10b + a$

According to the problem, this new number exceeds the original number by 18. So, we can write the equation:

$10b + a = (10a + b) + 18$

Simplifying this equation:

$10b + a = 10a + b + 18$

Now, subtract $b$ and $a$ from both sides:

$9b - 9a = 18$

Dividing through by 9:

b - a = 2 \tag{2}

Step 3: Solve the system of equations

Now, we have the following system of equations:

1. $a + b = 8$
2. $b - a = 2$

We can solve these equations by substitution or addition.

From equation (2), solve for $b$:

$b = a + 2$

Substitute this expression for $b$ into equation (1):

$a + (a + 2) = 8$

Simplifying:

$2a + 2 = 8$

Subtract 2 from both sides:

$2a = 6$

Divide by 2:

$a = 3$

Step 4: Find $b$

Substitute $a = 3$ into the equation $b = a + 2$:

$b = 3 + 2 = 5$

Step 5: Find the original number

The original number is:

$10a + b = 10(3) + 5 = 35$

Thus, the original number is 35.

Verification:

• The sum of the digits is $3 + 5 = 8$, which satisfies the first condition.
• Interchanging the digits gives $53$, and $53 - 35 = 18$, which satisfies the second condition.

So, the original number is 35.

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Would you like a more detailed breakdown of any part of the solution? Or do you have any other questions?

Here are some related questions for further exploration:

1. How can we solve for a three-digit number where the sum of the digits is 15?
2. What happens if the sum of the digits of a number is a negative value?
3. How would you approach a problem where the difference between two numbers formed by digit interchange is given?
4. How can you solve similar problems using algebraic expressions for three-digit or higher numbers?
5. How do you handle problems where the sum or difference of digits is not an integer?

Tip: When solving problems involving digit manipulation, it's helpful to express the number as $10a + b$ and break the problem down into simpler equations based on the relationships between the digits.