There are certain 2-digit numbers. The difference between the number and the one obtained on reversing it is always 27. How many such maximum 2-digit numbers are there?
- (a) 3
- (b) 4
- (c) 5
- (d) None of the above ✓ UPSC's answer
Why the answer is (d)
• Let the 2-digit number be 10a + b (a ∈ {1,…,9}, b ∈ {0,…,9}); its reverse is 10b + a, and the difference is (10a + b) − (10b + a) = 9(a − b).
• Setting |9(a − b)| = 27 gives |a − b| = 3, so either a − b = 3 or b − a = 3.
• When a − b = 3 the valid pairs (a, b) are (3,0), (4,1), (5,2), (6,3), (7,4), (8,5), (9,6), yielding the numbers 30, 41, 52, 63, 74, 85, 96 — seven numbers.
• When b − a = 3 the valid pairs (a, b) are (1,4), (2,5), (3,6), (4,7), (5,8), (6,9), yielding the numbers 14, 25, 36, 47, 58, 69 — six numbers.
• In total there are 7 + 6 = 13 two-digit numbers whose difference from their reverse is 27, which is none of 3, 4, or 5, so the correct choice is (d) None of the above.
Why the other options are wrong
- (a) 3
- The count 3 would arise only if one mistakenly restricted the digit range (e.g., allowing a − b = 3 with a ≤ 5), but the full range a = 3 to 9 and a = 1 to 6 gives 13 numbers, not 3.
- (b) 4
- The count 4 does not correspond to any consistent restriction on the digits; the equation |a − b| = 3 is satisfied by 13 distinct two-digit numbers, not 4.
- (c) 5
- The count 5 is not supported by the arithmetic: solving |a − b| = 3 over all valid digit pairs yields 13 numbers, so 5 undercounts the solution set.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2017, held on 18 June 2017. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.