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?
- (a) 2
- (b) 4
- (c) 6
- (d) 8 ✓ UPSC's answer
Why the answer is (d)
• Write X = 10a + b; then Y = 10b + a, and X + Y = 11(a + b).
• The greatest two-digit number is 99, so a + b = 9.
• Both X and Y must be two-digit numbers, so neither digit can be 0 — this rules out 90 (whose reverse, 09, is not two-digit).
• The possible values are 18, 27, 36, 45, 54, 63, 72 and 81 — eight values, option (d).
Why the other options are wrong
- (a) 2
- Option (a) is incorrect because there are 8 valid pairs of digits summing to 9, not 2.
- (b) 4
- Option (b) is incorrect because it undercounts the valid digit pairs; there are 8, not 4.
- (c) 6
- Option (c) is incorrect because it misses two valid pairs; the total count is 8, not 6.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.