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A four-digit odd number A has a unit digit that is not 5. When divided by a prime number, it gives a three-digit number B whose tens digit is 0. The sum of the digits of A is 8, while that of B is 5. Which of the following statements is/are correct about A and B? I. The product of the digits of A is equal to the sum of its digits. II. The difference between the hundreds digit of A and B is 3. III. The sum of A and B is 1624.

1. Only (II)
2. All of the above
3. Only (III)
4. Only (I)
5. Only (I) and (III)

Correct answer: Only (II)

Solution

The digit-sum and divisibility conditions uniquely determine the numbers, and only statement II matches the resulting values. Statements I and III do not hold for the identified pair.

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