Correct Option (A)
The given Identity Card number is ABCDEFG, where each letter represents a distinct digit from the set {1, 2, 4, 5, 7, 8, 9}.
Applying the divisibility rules:
- The 7-digit number ABCDEFG is divisible by 9. The sum of all given digits is 1 + 2 + 4 + 5 + 7 + 8 + 9 = 36, which is divisible by 9. This condition is inherently satisfied by the set of digits.
- After deleting the first digit from the right (G), the resulting number ABCDEF is divisible by 6. This implies F must be an even digit, and the sum A+B+C+D+E+F must be divisible by 3.
- After deleting two digits from the right (G, F), the resulting number ABCDE is divisible by 5. Since 0 is not among the allowed digits, the digit E must be 5.
- After deleting three digits from the right (G, F, E), the resulting number ABCD is divisible by 4. This implies that the number formed by the last two digits, CD, must be divisible by 4. For CD to be divisible by 4, D must be an even digit.
- After deleting four digits from the right (G, F, E, D), the resulting number ABC is divisible by 3. This implies that the sum A+B+C must be divisible by 3.
- After deleting five digits from the right (G, F, E, D, C), the resulting number AB is divisible by 2. This implies B must be an even digit.
From these deductions:
- E = 5 (an odd digit).
- B, D, F are even digits. The available even digits are {2, 4, 8}. Therefore, {B, D, F} = {2, 4, 8}.
- A, C, G are odd digits. The available odd digits (excluding E=5) are {1, 7, 9}. Therefore, {A, C, G} = {1, 7, 9}.
Consider the condition that CD is divisible by 4. C is an odd digit from {1, 7, 9}, and D is an even digit from {2, 4, 8}.
- If D = 2: Possible CD values are 12, 72, 92. All are divisible by 4.
- If D = 4: Possible CD values are 14, 74, 94. None are divisible by 4.
- If D = 8: Possible CD values are 18, 78, 98. None are divisible by 4.
Thus, D must be 2. The remaining even digits for B and F are {4, 8}.
We need to find the possible value for the sum of the middle three digits, which are C, D, and E.
We have E = 5 and D = 2. C is an odd digit from {1, 7, 9}.
The sum C + D + E = C + 2 + 5 = C + 7.
Possible values for C + D + E:
- If C = 1, then Sum = 1 + 2 + 5 = 8.
- If C = 7, then Sum = 7 + 2 + 5 = 14.
- If C = 9, then Sum = 9 + 2 + 5 = 16.
The possible sums for the middle three digits are 8, 14, and 16. Among the given options, 8 is a possible value.
Incorrect Options:
Options (B) 9, (C) 11, and (D) 12 are incorrect because, based on the derived properties, the sum of the middle three digits (C + D + E) must be an even number (Odd + Even + Odd = Even). Both 9 and 11 are odd numbers, making them impossible sums. While 12 is an even number, it is not one of the possible sums (8, 14, 16) calculated from the available distinct digits and divisibility rules.