A two-digit number has a digit sum of 14, and the absolute difference between its digits is 2. What is the product of the two digits of this two-digit number?

Introduction / Context:
This is a classic two-digit digit-algebra problem. You are given the sum and the absolute difference of the digits and asked to determine their product. Such problems measure skill in setting up and solving small linear systems for digits constrained between 0 and 9.

Given Data / Assumptions:

• Let the tens digit be a and the units digit be b.
• a + b = 14.
• |a - b| = 2.
• Digits are integers with 0 ≤ a, b ≤ 9 and a ≠ 0 (two-digit number).

Concept / Approach:
Assume without loss of generality a ≥ b, so a - b = 2. Solve the system with two equations in two unknowns, then compute the product a * b.

Step-by-Step Solution:

Verification / Alternative check:
Check the difference: |8 - 6| = 2 and sum: 8 + 6 = 14; both conditions satisfied.

Why Other Options Are Wrong:
56 corresponds to 7 * 8 (sum 15). 45 corresponds to 9 * 5 (difference 4). 36 corresponds to 9 * 4 (sum 13). “None of the above” is unnecessary because 48 fits.

Common Pitfalls:
Using the wrong sign for the difference or swapping digits without checking the absolute difference condition.

Final Answer:
48