Divisibility rules application to find a 7-digit identity card number
Question
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?
Options
8
9
11
12
Explanation
Use divisibility rules on the cascading string ABCDEFG. The available digits are {1, 2, 4, 5, 7, 8, 9}.
Since B, D, F must be the three even digits, they are {2, 4, 8} in some order. The odd digits A, C, G are {1, 7, 9}.
Since CD must be a multiple of 4: C is odd (1 or 7), and D is even.
Thus, D must be 2. The remaining even digits are B and F in {4, 8}.
Since ABCDEF is divisible by 3, and ABC is divisible by 3 (from the ABC divisibility rule), the sum D+E+F must be a multiple of 3. D+E+F = 2 + 5 + F = 7 + F. If F = 8, sum = 15 (valid). If F = 4, sum = 11 (invalid). Thus, F = 8 and B = 4.
The middle three digits are C, D, E. We know D = 2 and E = 5. Sum = C + 2 + 5 = C + 7. Since C is either 1 or 7:
For cascading divisibility cryptarithms, lock the 5-rule and the 2-rule slots first, then use the 4-rule (last two digits) to filter the even assignments. Answer: (a).
Question details
Year
2022
Paper
CSAT
Question
Q5
Section
Numerical Ability
Sub-topic
Divisibility Rules
Type
Factual single
Difficulty
Hard
Source hint
Identity Card ABCDEFG divisibility puzzle
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