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)

• Let the two-digit number X be represented as 10a + b, where a is the tens digit and b is the units digit.

• The number Y, formed by interchanging the digits, is 10b + a.

• The sum X + Y is (10a + b) + (10b + a) = 11a + 11b = 11(a + b).

• The greatest two-digit number is 99, so we set 11(a + b) = 99, which simplifies to a + b = 9.

• Since X is a two-digit number, the tens digit a can range from 1 to 9, and the units digit b can range from 0 to 9.

• The possible pairs (a, b) that sum to 9 are (1,8), (2,7), (3,6), (4,5), (5,4), (6,3), (7,2), and (8,1), giving exactly 8 possible values for X.

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.