A student can arrange her collection of books into equal groups of 5 books, 9 books, or 13 books each, with no book left over in any case. What is the smallest possible total number of books that the student can have?

Introduction / Context:
This aptitude question tests understanding of the Least Common Multiple, or LCM, which is a common concept in number system problems. When a quantity can be grouped in several different ways without leaving any remainder, it means the total quantity is exactly divisible by all the given group sizes. The smallest such total is the LCM of the given numbers. Understanding LCM is very useful in scheduling, grouping, and divisibility based questions that often appear in competitive exams.

Given Data / Assumptions:

• The student can arrange books in groups of 5.
• The student can arrange books in groups of 9.
• The student can arrange books in groups of 13.
• No book is left over in any of these arrangements.
• We need the smallest possible total number of books.

Concept / Approach:
To find the smallest positive number that is divisible by 5, 9, and 13, we calculate the LCM of these three numbers. The LCM is found by taking prime factorizations and choosing the highest power of each prime involved. The resulting LCM represents the smallest number that can be divided exactly by all the given divisors, which in this context corresponds to the smallest possible number of books.

Step-by-Step Solution:
Step 1: Prime factorize each number. 5 = 5 9 = 3 * 3 = 3^2 13 = 13 Step 2: Collect highest powers of primes. We have primes 3, 5, and 13. LCM = 3^2 * 5 * 13 Step 3: Multiply to get the LCM. 3^2 = 9 9 * 5 = 45 45 * 13 = 585 Therefore, the smallest total number of books is 585.

Verification / Alternative check:
We verify by dividing 585 by each group size. 585 / 5 = 117, 585 / 9 = 65, and 585 / 13 = 45. All quotients are integers, and there is no remainder in any case. This confirms that 585 is divisible by 5, 9, and 13, and hence satisfies the condition of forming equal groups for all three sizes. Because LCM gives the smallest such number, 585 is indeed the minimum possible total number of books.

Why Other Options Are Wrong:
487: Not divisible by 5 or 9, so it cannot form groups of those sizes without remainder.
635: Divisible by 5 but not divisible by 9 or 13 together, so it fails the requirement.
705: Divisible by 5 and 9 but not exactly by 13, so it is not valid for all group sizes.
845: Divisible by 5 and 13 but not by 9, hence it does not satisfy all the conditions.

Common Pitfalls:
Many students mistakenly try to test numbers one by one or simply multiply all three numbers without using LCM logic. Another common mistake is to confuse HCF (the greatest common divisor) with LCM. Some examinees also forget to include the highest power of a prime factor when finding the LCM, leading to an incorrect smaller value. Remember that LCM must be divisible by each number completely, without remainder.

Final Answer:
The smallest possible number of books is 585.