A four-digit odd number 'M' whose unit digit is not 5 when divided by a prime number gives resultant as a three-digit number 'N' whose tens place digit is 0. The sum of digits of M is 8 while that of N is 5. If M

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A four-digit odd number 'M' whose unit digit is not 5 when divided by a prime number gives resultant as a three-digit number 'N' whose tens place digit is 0. The sum of digits of M is 8 while that of N is 5. If M < 3400 then which of the following s…

Accepted Answer

Concept & Number Properties: This problem requires synthesizing multiple number properties: divisibility, prime numbers, digit sums, and parity (odd/even). Let M = N × p, where p is a prime number. Since M is odd, both factors N and p must strictly be odd numbers. Step-by-Step Solution: 1. Determine the possibilities for N: N is a 3-digit number. Tens place of N = 0. Sum of digits of N = 5. Possible configurations for N: 104, 203, 302, 401. Since M is odd, N must be odd. This eliminates 104 and 302. N is either 203 or 401. 2. Test prime multipliers for N = 401: M = 401 × p (where p is an odd prime, so p \in \{3, 5, 7, \dots\}) - If p=3 \implies M = 1203. Sum of digits = 1+2+0+3 = 6 (Does not equal 8). - If p=5 \implies M = 2005. Unit digit is 5 (Not allowed by the rules). - If p=7 \implies M = 2807. Sum of digits = 2+8+0+7 = 17 (Does not equal 8). 3. Test prime multipliers for N = 203: M = 203 × p - If p=3 \implies M = 609 (Not a 4-digit number). - If p=5 \implies M = 1015. Unit digit is 5 (Not allowed). - If p=7 \implies M = 1421. Sum of digits = 1+4+2+1 = 8. Unit digit is 1 (not 5). M < 3400. This perfectly satisfies all conditions! So, M = 1421 and N = 203. 4. Evaluate the statements: - Statement I: Product of digits of M = 1 × 4 × 2 × 1 = 8. Sum of digits of N = 2 + 0 + 3 = 5. Is 8 three more than 5? Yes (8 = 5 + 3). (TRUE) - Statement II: Hundred place digit of M is 4. Hundred place digit of N is 2. Difference = 4 - 2 = 2. (TRUE) - Statement III: Sum of M and N = 1421 + 203 = 1624. This is not 1729. (FALSE) Statements I and II are correct. Exam Strategy & Shortcut: Limit your search space immediately. The condition "Tens digit is 0" and "Sum is 5" restricted N to only four numbers instantly. Identifying that N MUST be odd halves the work. Always process the most restrictive constraints first to rapidly reach the correct numbers. Common Pitfall: A frequent error is forgetting that if a product (M) is odd, ALL of its integer factors must be odd. Students might waste time testing even numbers for N or p. Final Answer: Therefore, the correct answer is Only (I) and (II).

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