Directions: For a 5-digit number, without repetition of digits, the following information is available. (i) The first digit is more than 5 times the last digit. (ii) The two-digit number formed by the last two digits is the product of two prime numbers. (iii) The first three digits are all odd. (iv) The number does not contain the digits 3 or 0 and the first digit is also the largest. Which of the following is a factor of the given number?

### Concept & Logic To determine the factors of the number, we first need to deduce all five digits. Once the number is known, we can apply standard divisibility rules (like the sum of digits for $3$ and $9$) to check the options. ### Step-by-Step Solution Let the 5-digit number be $A B C D E$. From previous deductions: 1. $E = 1$ (because $A > 5E$ and $0$ is not allowed). 2. $\{A, B, C\} = \{5, 7, 9\}$ (because they are odd, distinct, and $1$ is used). 3. $A = 9$ (because it's the largest). So, $B$ and $C$ are $5$ and $7$ in some order. Now we must find $D$. The remaining available even digits are $2, 4, 6, 8$. Statement (ii) says the two-digit number $D E$ is the product of two prime numbers. Since $E = 1$, the number is $D1$. Let's test the possibilities: * If $D = 2 \implies 21 = 3 \times 7$ (Both $3$ and $7$ are primes. Valid.) * If $D = 4 \implies 41$ (Prime number, not a product. Invalid.) * If $D = 6 \implies 61$ (Prime number. Invalid.) * If $D = 8 \implies 81 = 3 \times 3 \times 3 \times 3$ (Product of four primes. Invalid.) Thus, $D$ must be $2$. The 5-digit number is either $95721$ or $97521$. Now, check the given options for factors: * **Check 2 and 4:** The number ends in $1$ (an odd number), so it is not divisible by $2$ or $4$. * **Check 3:** The divisibility rule for $3$ states the sum of digits must be a multiple of $3$. Sum = $9 + 5 + 7 + 2 + 1 = 24$. Since $24$ is divisible by $3$, the number is divisible by $3$. * **Check 9:** The sum is $24$, which is not divisible by $9$. ### Exam Strategy & Shortcut Once you deduce that the sum of the five digits is always $9 + 5 + 7 + 2 + 1 = 24$, regardless of the internal order of $5$ and $7$, you can immediately apply the divisibility rule for $3$. A sum of $24$ guarantees divisibility by $3$ but eliminates $9$. ### Common Pitfall A common error is misinterpreting "product of two prime numbers" in clue (ii). Some students might test $81$ thinking $9 \times 9$, but $9$ is not prime. Being strict with the definition of prime numbers ($2, 3, 5, 7, 11...$) is crucial for deducing $D=2$. ### Final Answer Therefore, the correct answer is 3.