Find the two-digit number — which statements suffice? I. The difference between the number and its digit-reversed form is 27. II. The difference between the two digits is 3. III. The units digit is less than the tens digit by 3.

Introduction / Context:We must decide which statements allow us to identify a unique two-digit number. Represent the number as 10a + b with digits a (tens) and b (units).

Given Data / Assumptions:

• I: |(10a + b) − (10b + a)| = 27 ⇒ 9 * |a − b| = 27 ⇒ |a − b| = 3.
• II: |a − b| = 3.
• III: b = a − 3 (units less than tens by 3).

Concept / Approach:Combine conditions and see if a unique (a, b) pair emerges. With |a − b| = 3 and 1 ≤ a ≤ 9, 0 ≤ b ≤ 9, many numbers satisfy the constraints.

Step-by-Step Solution:

Verification / Alternative check:Test any candidate from the set; each meets I, II, III and yields the same 27 difference from its reverse. Non-uniqueness persists.

Why Other Options Are Wrong:

• Only I and II: Merely restate |a − b| = 3 ⇒ many solutions.
• Only I and III: Still produces a family of solutions with a − b = 3.
• Only I and either II or III: Same issue; no uniqueness.
• All I, II and III: Still not unique.

Common Pitfalls:Assuming the difference with reverse fixes the number; it fixes only the digit gap.

Final Answer:None of these