A two-digit number has digits that satisfy three conditions: (A) the sum of the digits is 15, (B) the difference of the squares of the digits is 45 (larger digit squared minus smaller digit squared), and (C) the difference of the digits (tens digit minus units digit) is 3. Which information is sufficient to determine the number?

Introduction / Context:
This is a data sufficiency style question about a two-digit number whose tens and units digits must satisfy several conditions. Rather than asking directly for the number, the problem asks which combination of statements provides enough information to determine the number uniquely. Understanding how to translate each condition into an equation and how they interact is key to solving data sufficiency questions correctly.

Given Data / Assumptions:
- Let the two-digit number have tens digit t and units digit u, with t between 1 and 9 and u between 0 and 9.
- Statement A: t + u = 15 (sum of the digits is 15).
- Statement B: t^2 - u^2 = 45 (difference of squares of the digits, larger digit squared minus smaller digit squared, equals 45).
- Statement C: t - u = 3 (difference of their digits, interpreted as tens digit minus units digit, is 3).
- The goal is to decide which pairs of statements are sufficient to find the unique number, not necessarily to compute it in every case, though computing it once can help verify sufficiency.

Concept / Approach:
We treat each statement as an equation and examine combinations. From B, we can factor t^2 - u^2 as (t - u)(t + u) = 45. Using integer factor pairs of 45, together with digit constraints, gives possible (t, u) pairs. We then see how combining A, B, and C in different pairs restricts the possibilities to a single number. If any pair of statements always leads to one unique (t, u) pair, that pair is sufficient.

Step-by-Step Solution:
Step 1: Use statement B alone: (t - u)(t + u) = 45. Possible positive factor pairs of 45 are (1, 45), (3, 15), and (5, 9). They give potential (t - u, t + u) pairs. Solving:
For (3, 15): t = (15 + 3) / 2 = 9, u = (15 - 3) / 2 = 6.
For (5, 9): t = (9 + 5) / 2 = 7, u = (9 - 5) / 2 = 2.
The pair (1, 45) would give t = 23, u = 22, which is invalid for digits. Step 2: From B alone we have two digit pairs: (9, 6) and (7, 2), corresponding to numbers 96 and 72, so B alone is not sufficient. Step 3: Combine A and B. A says t + u = 15. Among the pairs from B, only (9, 6) has sum 15 (7 + 2 = 9). Thus A and B together uniquely give t = 9 and u = 6, so the number is 96. Hence A and B together are sufficient. Step 4: Combine B and C. C says t - u = 3. Of the B-pairs (9, 6) and (7, 2), only (9, 6) has t - u = 3 (7 - 2 = 5). Therefore B and C together again uniquely give 96 and are sufficient. Step 5: Combine A and C. A gives t + u = 15, and C gives t - u = 3. Solving these simultaneously: adding the equations gives 2t = 18, so t = 9 and u = 6. This is a unique solution, so A and C together are also sufficient. Step 6: Since any one of the three pairs (A, B), (B, C), or (C, A) is sufficient by itself, the correct description is that any one of these pairs is enough.

Verification / Alternative check:
We can verify by checking that no other digit pair meets all conditions simultaneously. The pair (9, 6) satisfies t + u = 15, t^2 - u^2 = 81 - 36 = 45, and t - u = 3. The pair (7, 2) satisfies B but violates both A and C. Thus, each combination A & B, B & C, and A & C isolates the correct pair (9, 6) uniquely, confirming the sufficiency of each pair.

Why Other Options Are Wrong:
Option B and C together are sufficient: This is true but incomplete, because A and B together, and A and C together are also sufficient; the option fails to state the broader sufficiency pattern.
Option C and A together are sufficient: This is true but again does not acknowledge that B can pair with either A or C to yield sufficiency.
Option A and B together are sufficient: Also true but not the only sufficient pair.
Option None of these: Incorrect because a precise description exists, namely that any one of the three pairs is sufficient.

Common Pitfalls:
A common mistake is to treat data sufficiency as if only one specific pair must work, overlooking that other combinations may also be enough. Another pitfall is misinterpreting difference of the digits and difference of squares as absolute differences rather than oriented differences, which can cause ambiguity in solving. Clearly specifying t - u and t^2 - u^2 as oriented differences ensures consistent equations and a unique solution in each pair case.

Final Answer:
Any one pair of A and B, B and C, or C and A is sufficient to determine the number uniquely.