Number System — A two-digit number has digit sum 10. When its digits are reversed, the new number is greater than the original by 72. Determine the original two-digit number.

Introduction / Context:
This is a classic digits problem involving a two-digit number, relationships between the tens and ones digits, and how reversing digits changes the value. We translate the words into equations using basic place-value reasoning to solve efficiently without trial-and-error.

Given Data / Assumptions:

• The number is two-digit with tens digit a and units digit b.
• Digit sum condition: a + b = 10.
• Reversal condition: reversing digits increases the number by 72.
• All digits are integers from 0 to 9, with a ≥ 1 for a two-digit number.

Concept / Approach:
Represent the original number as 10*a + b and the reversed number as 10*b + a. The sentence “increases by 72” gives a linear equation relating a and b. Solve the system composed of the sum condition and the reversal equation to get exact digits. This avoids guesswork and guarantees the correct two-digit number.

Step-by-Step Solution:
1) Let original = 10*a + b; reversed = 10*b + a.2) From the prompt: (10*b + a) - (10*a + b) = 72.3) Simplify: 9*b - 9*a = 72 → b - a = 8.4) Combine with a + b = 10. Add the equations: (b - a) + (a + b) = 8 + 10 → 2*b = 18 → b = 9.5) Substitute back: a + 9 = 10 → a = 1. Original number = 10*a + b = 10*1 + 9 = 19.

Verification / Alternative check:
Check both conditions: sum of digits 1 + 9 = 10 (satisfied). Reversed number is 91; difference 91 - 19 = 72 (satisfied). Thus 19 is correct.

Why Other Options Are Wrong:
64: digits sum to 10 but reversed 46 differs by -18, not +72. 82: sum is 10 but 28 - 82 = -54, not +72. 37: sum is 10 but 73 - 37 = 36, not 72. 46: sum is 10 but 64 - 46 = 18, not 72.

Common Pitfalls:
Mixing up which number is “original” vs “reversed”; solving a + b = 10 but forgetting to enforce the 72-difference condition; treating 0 as a valid tens digit (which would not be two-digit).

Final Answer:
19