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. The last digit of the number is

### Concept & Logic This problem tests your ability to translate algebraic inequalities into discrete single-digit values. The key insight lies in leveraging the maximum possible value of a digit to constrain the minimum value of another. ### Step-by-Step Solution Let the 5-digit number be $A B C D E$. Statement (iv) states that the digits $3$ and $0$ are not allowed in the number. Statement (i) provides the inequality: $A > 5E$. Because $A$ is a single digit, its maximum possible value is $9$. Substitute this maximum value into the inequality: $$9 > 5E$$ $$E < 1.8$$ Since $E$ is a digit, it must be an integer. The only non-negative integers less than $1.8$ are $0$ and $1$. However, according to statement (iv), the digit $0$ is not permitted in this number. Therefore, the only possible valid digit for $E$ (the last digit) is $1$. ### Exam Strategy & Shortcut Focus purely on the clues mentioning the target variable. You are asked for the last digit ($E$). Statement (i) connects the first digit to the last digit with a $5 \times$ multiplier. Because digits max out at $9$, $5 \times 2 = 10$ is already too large. Thus, the last digit must be $1$ (since $0$ is banned). You can solve this in 5 seconds without reading the other clues. ### Common Pitfall The most common mistake is forgetting that $0$ is explicitly excluded in clue (iv), leading some students to think $E$ could be $0$ or $1$ and resulting in confusion. Always keep explicit exclusions in mind. ### Final Answer Therefore, the correct answer is 1.