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 statement/s is/are correct about M and N? I. Product of the digits of M is 3 more than the sum of the digits of N II. Difference between the hundred place digits of M and N is 2 III. Sum of M and N is 1729

### Concept & Number Properties This problem requires synthesizing multiple number properties: divisibility, prime numbers, digit sums, and parity (odd/even). Let $M = N \times p$, 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 $N = 401$:** $M = 401 \times p$ (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 $N = 203$:** $M = 203 \times p$ - 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 $M = 1 \times 4 \times 2 \times 1 = 8$. 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)**.