Question

1. a rectangle measures 72 cm by 108 cm. a) a 2 cm by 2 cm square can be used to cover the rectangle without any spaces or overlapping. explain why. b) how can you use the common factors of 72 and 108 to identify all the squares that can be used to cover the rectangle? c) list all other squares with whole - number dimensions that can be used to cover the rectangle.

Explanation:

Step1: Check divisibility for part a

To cover a rectangle with a square without spaces or overlapping, the side - length of the square must divide both the length and width of the rectangle evenly. For a 2 - cm square and a rectangle of dimensions 72 cm by 108 cm, we check if 72 and 108 are divisible by 2. Since \(72\div2 = 36\) and \(108\div2=54\), the 2 - cm square can cover the rectangle without spaces or overlapping.

Step2: Understand the concept for part b

The side - length of a square that can cover the rectangle without spaces or overlapping must be a common factor of the length and width of the rectangle. If \(l\) is the length and \(w\) is the width of the rectangle, and \(s\) is the side - length of the square, then \(l = m\times s\) and \(w=n\times s\) for some non - negative integers \(m\) and \(n\). So, the common factors of 72 and 108 give the possible side - lengths of the squares.

Step3: Find common factors for part c

First, find the factors of 72: \(72=1\times72,2\times36,3\times24,4\times18,6\times12,8\times9\), so the factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Next, find the factors of 108: \(108 = 1\times108,2\times54,3\times36,4\times27,6\times18,9\times12\), so the factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108.
The common factors of 72 and 108 are 1, 2, 3, 4, 6, 9, 12, 18, 36. We already know about the 2 - cm square. The other squares are 1 cm by 1 cm, 3 cm by 3 cm, 4 cm by 4 cm, 6 cm by 6 cm, 9 cm by 9 cm, 12 cm by 12 cm, 18 cm by 18 cm, 36 cm by 36 cm.

Answer:

a) Since 72 and 108 are divisible by 2 (\(72\div2 = 36\) and \(108\div2 = 54\)), a 2 - cm square can cover the rectangle without spaces or overlapping.
b) The side - length of a square that can cover the rectangle without spaces or overlapping must be a common factor of the length and width of the rectangle.
c) 1 cm by 1 cm, 3 cm by 3 cm, 4 cm by 4 cm, 6 cm by 6 cm, 9 cm by 9 cm, 12 cm by 12 cm, 18 cm by 18 cm, 36 cm by 36 cm.