Sophia has an unlimited supply of square tiles. Sophia has \(1\text{ cm}\) by \(1\text{ cm}\) tiles, \(2 \text{ cm}\) by \(2\text{ cm}\) tiles, \(3\text{ cm}\) by \(3 \text{ cm}\) tiles, and so on. Every tile has integer side lengths, in centimetres.
A rectangular bathroom floor with an \(84 \text{ cm}\) by \(140 \text{ cm}\) surface is to be completely covered by identical square tiles, none of which can be cut. Sophia knows that the floor can be completely covered with \(11\,760\) tiles of size \(1\text{ cm}\) by \(1 \text{ cm}\), since \(84\times 140=11\,760\text{ cm}^2.\) However, Sophia wants to use the minimum number of identical sized tiles to complete the job.
Determine the minimum number of identical sized tiles required to completely cover the bathroom floor.
To use the least number of tiles, Sophia must use the largest tile possible. The square tile must have sides less than or equal to \(84\text{ cm}\). If the tile has side length greater than \(84\text{ cm}\), it would have to be cut to fit the width of the floor.
Since the tiles are square and must completely cover the floor, the side length of the tile must be a number that is a factor of both \(84\) and \(140\). In fact, since Sophia wants the largest side length, she is looking for the greatest common factor of \(84\) and \(140.\)
The positive factors of \(84\) are \(1\), \(2\), \(3\), \(4\), \(6\), \(7\), \(12\), \(14\), \(21\), \(28\), \(42\), and \(84.\)
The positive factors of \(140\) are \(1\), \(2\), \(4\), \(5\), \(7\), \(10\), \(14\), \(20\), \(28\), \(35\), \(70\), and \(140.\)
The largest number common to both lists is \(28.\) Therefore, the greatest common factor of \(84\) and \(140\) is \(28\). The required tiles are \(28\text{ cm}\) by \(28\text{ cm}\). Since \(84\div 28=3\), the surface is \(3\) tiles wide. Since \(140\div 28=5\), the surface is \(5\) tiles long. The minimum number of tiles required is \(3\times 5=15\) tiles.
The number of \(28\text{ cm}\) by \(28\text{ cm}\) tiles required to cover the floor area is \(15\). This is the minimum number of tiles required.