Ammini is arranging 36 identical squares in the form of different rectangles.…

Concept: When identical unit squares are joined edge-to-edge to form a rectangle, the rectangle's two side lengths must be a pair of positive integers whose product equals the total number of squares. Because a rectangle with sides a and b is the same shape as one with sides b and a, the number of distinct rectangles possible equals the number of unordered factor pairs (divisor pairs) of that total — every divisor pairs with a complementary divisor to give one rectangle.

1. List every factor (divisor) of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine factors in all.

2. Pair each factor with its complementary factor so that the two multiply to 36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6).

3. Count the pairs formed above — each pair is one distinct rectangle shape, giving five such pairs.

Cross-check: multiply out every pair to confirm each equals 36 — 1×36, 2×18, 3×12, 4×9 and 6×6 all give 36, and since 36 = 6² is a perfect square, the pair (6, 6) contributes only once (a square arrangement is still a rectangle, but it is not counted twice).

Hence, Ammini can arrange the 36 squares into exactly five different types of rectangles.