Let X be a two-digit number and Y be another two-digit number formed by interchanging the digits of X. If (X + Y) is the greatest two-digit number, then what is the number of possible values of X?
Answer & explanation
Answer: (d) 8
X + Y = 11 × (sum of the digits), and it must equal 99, so the digits add up to 9. Both numbers must be two-digit, so neither digit can be 0: X can be 18, 27, 36, 45, 54, 63, 72 or 81, which is 8 values.
- Let X = 10a + b and Y = 10b + a; then X + Y = 11(a + b).
- The greatest two-digit number is 99, so 11(a + b) = 99 and a + b = 9.
- Y must also be a two-digit number, so b cannot be 0; this rules out X = 90 (its reverse is 09).
- a = 1 to 8 gives X = 18, 27, 36, 45, 54, 63, 72, 81, i.e. 8 values.
Remember · A number plus its reverse is 11 × digit sum. Watch for a 0 digit that makes the reverse a one-digit number.
Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·