Numbers from square-difference facts: The difference of two numbers is 3, and the difference of their squares is 39. Find the larger number.

Introduction / Context:
Using the identity a^2 − b^2 = (a − b)(a + b) allows you to relate differences of numbers to differences of their squares. This question checks whether you can translate conditions into simultaneous equations and solve quickly.

Given Data / Assumptions:

• Let the numbers be a (larger) and b (smaller).
• a − b = 3.
• a^2 − b^2 = 39.

Concept / Approach:
Apply a^2 − b^2 = (a − b)(a + b). Since a − b is given, you can immediately solve for a + b, then solve for a and b by adding and subtracting the two linear equations.

Step-by-Step Solution:

Verification / Alternative check:
Check the conditions: difference 8 − 5 = 3; square difference 64 − 25 = 39. Both match.

Why Other Options Are Wrong:
9, 12, 13, and 11 do not satisfy both equations simultaneously when paired with the corresponding b and thus violate either the difference or the square difference.

Common Pitfalls:
Confusing a^2 − b^2 with (a − b)^2, or trying to guess numbers without using the identity leads to wasted time and errors.

Final Answer:
8