A four-digit odd number *A* whose unit digit is not 5 and when divided by a prime number gives resultant as a three-digit number *B* whose tens place is 0. Sum of digits of *A* is 8 while that of *B* is 5. Which of the following statement/s is/are correct about *A* and *B*?
I. Product of the digits of *A* is equal to its sum of the digits
II. Difference between the hundred place digit of *A* and *B* is 3
III. Sum of *A* and *B* is 1624

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**Solution:**

1. Given *A* is a four-digit odd number, unit digit not 5, sum of digits of *A* is 8.
2. When *A* is divided by a prime number, the quotient is a three-digit number *B* whose tens digit is 0 and sum of digits of *B* is 5.

3. Possible values of *B* (three-digit, tens digit 0, sum of digits 5): 104, 203, 302, 401, 500.

4. Multiplying these by primes to get *A* (four-digit odd number with digit sum 8):
   - 203 × 7 = 1421 (sum digits: 1+4+2+1=8, odd number)
   - 401 × 3 = 1203 (sum digits: 1+2+0+3=6, not 8)

5. So, *A* = 1421 and *B* = 203.

Check statements:

- (I) Product of digits of *A* = 1 × 4 × 2 × 1 = 8, sum of digits = 8 → True
- (II) Difference between hundred place digits: *A* hundred digit = 4, *B* hundred digit = 2, difference = 2 ≠ 3 → False
- (III) Sum of *A* and *B* = 1421 + 203 = 1624 → True

**Answer:** Only (I) and (III) are correct, option (e).

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