Two-digit number with digit-sum and digit-swap conditions Interchanging the digits makes the number 27 larger; the sum of the digits is 13. What is the original two-digit number?

Introduction / Context:
Problems with two-digit numbers often use digit variables for tens and units. Here, the sum of digits and the effect of swapping digits are given, allowing a clean system of linear equations in the digit variables.

Given Data / Assumptions:

• Let the tens digit be t and the units digit be u.
• Original number = 10t + u.
• Swapped number = 10u + t.
• 10u + t = (10t + u) + 27.
• t + u = 13.

Concept / Approach:
Use the swap relation and the sum relation. Rearrange to isolate u − t, then combine with t + u to find the unique pair (t, u). Ensure digits are integers from 0 to 9 and tens digit is nonzero.

Step-by-Step Solution:
From swap relation: 10u + t = 10t + u + 27 ⇒ 9u − 9t = 27 ⇒ u − t = 3.Sum relation: t + u = 13.Add both equations: (u − t) + (t + u) = 3 + 13 ⇒ 2u = 16 ⇒ u = 8.Then t = 13 − u = 5.Original number = 10t + u = 10*5 + 8 = 58.

Verification / Alternative check:
Swap digits to get 85. Difference 85 − 58 = 27. Sum of digits 5 + 8 = 13. Both conditions hold.

Why Other Options Are Wrong:
67, 76, 85: Each fails either the digit-sum condition, the swap-difference condition, or both (note 85 is the swapped number, not the original).

Common Pitfalls:
Mixing the swap direction (thinking original is larger); or writing 10t + u and 10u + t incorrectly. Keep consistent definitions for tens and units throughout.

Final Answer:
58