Certain 3-digit numbers have the following characteristics : 1. All the three digits are different. 2. The number is divisible by 7. 3. The number on reversing the digits is also divisible by 7. How many such 3-digit numbers are there?
- (a) 2
- (b) 4 ✓ UPSC's answer
- (c) 6
- (d) 8
Why the answer is (b)
• Let the number be 100a + 10b + c and its reverse be 100c + 10b + a; since both are divisible by 7, their difference 99(a - c) is divisible by 7.
• Since 99 ≡ 1 (mod 7), a - c must be a multiple of 7; with different digits and a nonzero, the possible (a, c) pairs are (7,0), (8,1), (9,2), (1,8), and (2,9).
• For each pair, impose 100a + 10b + c ≡ 0 (mod 7), which simplifies to 2a + 3b + c ≡ 0 (mod 7), and choose b distinct from a and c.
• This gives b = 6 for (8,1) and (1,8), b = 5 for (9,2) and (2,9), while (7,0) forces b = 0 or 7, both invalid.
• Hence the numbers are 861, 168, 952, and 259, so there are 4 such numbers, matching option (b).
Why the other options are wrong
- (a) 2
- It is too low because it counts only the pair 168 and 861 and omits the valid pair 259 and 952.
- (c) 6
- It is too high because the (7,0) first-last digit pair gives middle digit 0 or 7, both violating the distinct-digit condition.
- (d) 8
- It is too high because only four of the possible digit arrangements satisfy the modulo-7 condition with all digits different.
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.