An Identity Card has the number ABCDEFG, not necessarily in that order, where each letter represents a distinct digit (1, 2, 4, 5, 7, 8, 9 only). The number is divisible by 9. After deleting the first digit from the right, the resulting number is divisible by 6. After deleting two digits from the right of original number, the resulting number is divisible by 5. After deleting three digits from the right of original number, the resulting number is divisible by 4. After deleting four digits from the right of original number, the resulting number is divisible by 3. After deleting five digits from the right of original number, the resulting number is divisible by 2. Which of the following is a possible value for the sum of the middle three digits of the number?
Answer & explanation
Answer: (a) 8
Work through the conditions: G must be 9, E must be 5, the even digits 2, 4, 8 fill B, D, F, and the tests for 3 and 4 fix B = 4 and D = 2. The middle digits C, D, E then add up to 8 (or 14, which is not offered).
- The digits 1, 2, 4, 5, 7, 8, 9 add up to 36, so the full number is divisible by 9 in any order.
- ABCDEF is divisible by 6, so F is even and A + B + C + D + E + F is a multiple of 3. That sum is 36 − G, so G is a multiple of 3: G = 9.
- ABCDE is divisible by 5 and 0 is not available, so E = 5.
- AB divisible by 2 makes B even; ABCD divisible by 4 makes D even. So B, D, F are the even digits 2, 4, 8, and A, C are 1 and 7 in some order.
- ABC divisible by 3: A + B + C = 1 + 7 + B = 8 + B must be a multiple of 3, so B = 4.
- ABCD divisible by 4: the last two digits CD must be 12 or 72 (18 and 78 fail), so D = 2 and F = 8.
- Middle three digits C + D + E = C + 2 + 5: this is 8 when C = 1 and 14 when C = 7. Only 8 is offered.
- Check: 7412589 — 741258 ÷ 6 = 123543, 74125 ends in 5, 7412 ÷ 4 = 1853, 741 ÷ 3 = 247, 74 is even.
Remember · Divisibility chains: fix the digit forced by 5, then the even places, then use digit-sum and last-two-digit tests.
Question and answer: UPSC's official GS Paper II (2022, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·