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Sasha arranges identical square tiles to form rectangles using the following rules:
1. Tiles must line up exactly without any gaps or overlaps.
2. The width of each rectangle must be larger than its height.
(a) Draw all the rectangles Sasha can form with 12 square tiles.
(b) Draw all the rectangles Sasha can form with 20 square tiles.
(c) Draw all the rectangles Sasha can form with 13 square tiles.
(d) Can Sasha form more rectangles with 20 square tiles or 28 square tiles? Justify your answer.
(e) Challenge: Sasha has some number of tiles less than 50 and is able to form only one rectangle. What is the largest number of tiles that Sasha could have?
(a) With 12 tiles, Sasha can form three rectangles:
– 12 × 1
– 6 × 2
– 4 × 3
(b) With 20 tiles, Sasha can form two rectangles:
– 20 × 1
– 10 × 2
(c) With 13 tiles, Sasha can form only one rectangle:
– 13 × 1
(d) Sasha can form more rectangles with 28 square tiles than with 20 square tiles.
For 20 tiles: 20 has factors 1, 2, 4, 5, 10, 20. This allows for 2 rectangles (as shown in part b).
For 28 tiles: 28 has factors 1, 2, 4, 7, 14, 28. This allows for 3 rectangles:
– 28 × 1
– 14 × 2
– 7 × 4
Therefore, Sasha can form more rectangles (3) with 28 tiles than with 20 tiles (2).
(e) The largest number of tiles less than 50 that can form only one rectangle would be a prime number. The largest prime number less than 50 is 47.
With 47 tiles, Sasha can only form a 47 × 1 rectangle.
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