A two-digit number equals three times the sum of its digits, and adding 45 to the number reverses its digits. What is the number?

Introduction / Context:
Puzzles about two-digit numbers typically translate into equations using the tens digit and units digit. Here we use two conditions: a value condition (three times the sum of digits) and a digit-reversal condition (adding 45 reverses the digits). Together they identify the number uniquely.

Given Data / Assumptions:

• Let the number be 10a + b, where a is the tens digit and b is the units digit.
• Condition 1: 10a + b = 3(a + b).
• Condition 2: (10a + b) + 45 = 10b + a (reversed digits).

Concept / Approach:
Convert both conditions into linear equations in a and b. Solve one to relate a and b, then use the other to confirm specific integer digits. Because digits are 0–9 and tens digit is 1–9, we can also check divisibility or parity constraints quickly.

Step-by-Step Solution:

Verification / Alternative check:
Confirm Condition 1 numerically: 27 = 3 * (2 + 7) = 3 * 9 = 27; satisfied. The reversal check also holds precisely.

Why Other Options Are Wrong:
32, 72, 23, and 54 do not satisfy both conditions simultaneously. For example, 72 is the reversed result after adding 45, not the original number.

Common Pitfalls:
Forgetting digits must be integers, or misapplying the reversal equation as subtracting 45 instead of adding.

Final Answer:
27