There are 36 identical books that must be arranged in rows so that each row contains the same number of books. Each row must have at least 3 books and there must be at least 3 rows. A row is defined as a line of books parallel to the front of the room. Under these conditions, how many different such arrangements are possible?

Introduction / Context:
This problem is about finding the number of ways to arrange identical objects into equal sized rows subject to minimum constraints on both the number of rows and the number of books in each row. Such questions are directly linked to the concept of factors and divisors of a number, and they frequently appear in basic number theory sections of aptitude tests.

Given Data / Assumptions:

• Total number of identical books: 36.
• The books are to be arranged in rows, each row having the same number of books.
• Each row must contain at least 3 books.
• There must be at least 3 rows.
• We count two arrangements as different if they have a different number of rows or a different number of books per row.

Concept / Approach:
Let the number of rows be r and the number of books in each row be k. The conditions imply r × k = 36, with r ≥ 3 and k ≥ 3. Thus we need to find all factor pairs (r, k) of 36 that satisfy these inequalities. Each valid factor pair corresponds to one distinct arrangement pattern, because the books are identical and only the grid structure matters.

Step-by-Step Solution:
Step 1: List all factor pairs of 36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6), (9, 4), (12, 3), (18, 2), (36, 1).Step 2: Apply the condition r ≥ 3 and k ≥ 3. This removes pairs where rows or books per row are less than 3.Step 3: Discard (1, 36), (2, 18), (18, 2) and (36, 1) because at least one of the numbers is less than 3.Step 4: The valid pairs that remain are (3, 12), (4, 9), (6, 6), (9, 4) and (12, 3).Step 5: Each of these five pairs corresponds to a distinct arrangement: 3 rows of 12, 4 rows of 9, 6 rows of 6, 9 rows of 4 and 12 rows of 3 books.Step 6: Therefore, the number of different arrangements that satisfy the conditions is 5.
Verification / Alternative check:
We can verify by explicitly checking the constraints for each valid pair. For (3, 12), we have 3 rows and 12 books per row, both at least 3. The same holds for (4, 9), (6, 6), (9, 4) and (12, 3). No other factor pair meets both minimum requirements, so we have found all possibilities. Hence the count 5 is confirmed.

Why Other Options Are Wrong:
Options 6, 7 or 8 would require more valid factor pairs than actually exist. Since 36 has only the listed factor pairs, and only 5 of them satisfy r ≥ 3 and k ≥ 3, any number greater than 5 must be incorrect. Option 4 is too small and would imply that one of the valid factor pairs had been missed or incorrectly eliminated.

Common Pitfalls:
Students sometimes forget that a pair like (3, 12) is distinct from (12, 3) only if the definition of rows versus columns is fixed. In this problem, the number of rows is the first coordinate, so both pairs are considered distinct valid arrangements. Another common error is to ignore the minimum conditions and count all factors without filtering, which would lead to 9 instead of 5.

Final Answer:
The number of different valid arrangements is 5.