The dimensions of a rectangular room are 12 feet 10 inches by 10 feet 1 inch. If square tiles of the same size are to cover the floor completely without any overlapping and if only whole tiles will be used, what is the largest possible length for a side of the tile?
To find the largest possible length for a side of the tile, we need to consider the factors or divisors of both the length and width of the room.
First, let's convert the dimensions to a consistent unit. Since we are dealing with inches, we'll convert the feet measurements to inches.
The length becomes 12 feet 10 inches, which is equal to (12 * 12) + 10 = 144 + 10 = 154 inches.
The width becomes 10 feet 1 inch, which is equal to (10 * 12) + 1 = 120 + 1 = 121 inches.
Now, let's find the factors of both 154 and 121.
The factors of 154 are 1, 2, 7, 11, 14, 22, 77, and 154.
The factors of 121 are 1 and 121.
Next, we need to find the greatest common divisor (GCD) of these two numbers, which is the largest factor shared by both numbers.
In this case, the GCD of 154 and 121 is 1 since it is the largest number that divides both 154 and 121 without leaving a remainder.
Therefore, the largest possible length for a side of the tile is 1 inch.