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?

Here's a step-by-step breakdown to solve the problem: 1. Available Digits: The letters A, B, C, D, E, F, G represent distinct digits from {1, 2, 4, 5, 7, 8, 9}. There are 7 digits and the number has 7 positions, so all digits are used exactly once. 2. Divisibility by 9 (ABCDEFG): A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the given digits (1+2+4+5+7+8+9) is 36, which is divisible by 9. This rule is always satisfied. 3. Divisibility by 5 (ABCDE): After deleting two digits from the right, ABCDE is divisible by 5. This means the last digit E must be 0 or 5. Since 0 is not in the available digits, E must be 5. So, E = 5. 4. Divisibility by 2 (AB): After deleting five digits from the right, AB is divisible by 2. This means B must be an even digit. So, B is from {2, 4, 8}. 5. Divisibility by 6 (ABCDEF): After deleting one digit from the right, ABCDEF is divisible by 6. This means F must be an even digit (for divisibility by 2) and the sum of its digits (A+B+C+D+E+F) must be divisible by 3. So, F is from {2, 4, 8}. Since B and F must be distinct digits, they are two different even digits from {2, 4, 8}. The third even digit from {2, 4, 8} must be D, A, C, or G. 6. Divisibility by 4 (ABCD): After deleting three digits from the right, ABCD is divisible by 4. This means the number formed by the last two digits, CD, must be divisible by 4. * We know E=5. B and F are distinct even digits from {2, 4, 8}. * Consider D: * If D is an odd digit (from {1, 7, 9}): For CD to be divisible by 4, C must be an even digit. The only available even digit for C would be the 'third even digit' (the one not used by B or F). Let's test possible C and D combinations: * If C=2: 21 (no), 27 (no), 29 (no). * If C=4: 41 (no), 47 (no), 49 (no). * If C=8: 81 (no), 87 (no), 89 (no). None of these result in a number divisible by 4. This means D cannot be odd. * Therefore, D must be the 'third even digit' from {2, 4, 8}. This implies that {B, F, D} = {2, 4, 8}. * Consequently, {A, C, G} must be the odd digits {1, 7, 9}. * Now, C is odd ({1, 7, 9}) and D is even ({2, 4, 8}). Let's re-check CD divisible by 4: * If D=2: C=1 => 12 (divisible by 4); C=7 => 72 (divisible by 4); C=9 => 92 (divisible by 4). * If D=4: C=1 => 14 (no); C=7 => 74 (no); C=9 => 94 (no). * If D=8: C=1 => 18 (no); C=7 => 78 (no); C=9 => 98 (no). * So, D must be 2. And C can be 1, 7, or 9. * Summary so far: E=5, D=2. {B, F} = {4, 8}. {A, C, G} = {1, 7, 9}. 7. Divisibility by 3 (ABC): After deleting four digits from the right, ABC is divisible by 3. This means A+B+C must be divisible by 3. Let's test the possible values for C: * Case C=1: * We have C=1, D=2, E=5. * {B, F} = {4, 8}. {A, G} = {7, 9}. * If B=4: A+B+C = A+4+1 = A+5. * If A=7: A+5 = 7+5 = 12 (divisible by 3). This is a valid partial sequence (A=7, B=4, C=1, D=2, E=5). * Remaining digits: F must be 8 (from {4,8}), G must be 9 (from {7,9}). * The full number is 7412589. * Let's check the divisibility by 6 for ABCDEF (741258): F=8 (even, so divisible by 2). Sum of digits (7+4+1+2+5+8 = 27) is divisible by 3. So 741258 is divisible by 6. This number is valid. * The middle three digits are C, D, E, which are 1, 2, 5. Their sum is 1+2+5 = 8. The sum of the middle three digits (C+D+E) is 8, which is option A. Since the question asks for a "possible value", and we found one, we can conclude. (For completeness, other combinations lead to invalid numbers or sums not in the options). The final answer is A) 8.