Find the two-digit number. A. Sum of the squares of the two digits of the…

Concept

This is a data-sufficiency problem. A statement (or a set of statements) is “sufficient” only if it narrows the unknown down to exactly ONE value. Write the number as 10t + u, where t is the tens digit (1–9) and u is the units digit (0–9). Each clue becomes an equation; we then count how many (t, u) pairs survive.

Translate each clue into an equation

• A: t2 + u2 = 26.

• B: (10t + u) : (t + u) = 5 : 2, i.e. 2(10t + u) = 5(t + u) → 15t = 3u → u = 5t.

• C: tens digit is 4 less than the units digit, i.e. t = u − 4.

Test each statement on its own

1. A alone: t2 + u2 = 26 is met by (1, 5) → 15 and (5, 1) → 51. Two numbers survive, so A alone is not sufficient.

2. B alone: u = 5t forces t = 1, u = 5 (any larger t pushes u past 9). Exactly one number, 15, survives, so B alone is sufficient.

3. C alone: t = u − 4 is met by (1,5), (2,6), (3,7), (4,8), (5,9) → 15, 26, 37, 48, 59. Five numbers survive, so C alone is not sufficient.

Combine the insufficient statements

A and C together: the pairs satisfying A are {(1,5), (5,1)}; of these, only (1,5) also satisfies t = u − 4. A single number, 15, survives, so A and C together are sufficient.

Cross-check

Every sufficient route lands on the same number, 15: from B, 15 : (1+5) = 15 : 6 = 5 : 2 ✓; and 15 satisfies A (12 + 52 = 1 + 25 = 26 ✓) together with C (1 = 5 − 4 ✓). So the number is fixed either by combining A with C, or by statement B on its own.