To find the smallest possible square that can be formed using the tiles measuring 12 centimeters by 6 centimeters without cutting or overlapping, we need to determine the least common multiple (LCM) of the dimensions of the tile.
The dimensions of the tile are:
- Length = 12 cm
- Width = 6 cm
To find the LCM, we can start by determining the prime factorization of each dimension:
-
For 12: \[ 12 = 2^2 \times 3^1 \]
-
For 6: \[ 6 = 2^1 \times 3^1 \]
The LCM is found by taking the highest power of each prime that occurs in the factorizations:
- For the prime number \(2\): The highest power is \(2^2\) (from 12).
- For the prime number \(3\): The highest power is \(3^1\) (common in both).
Thus, the LCM is: \[ LCM = 2^2 \times 3^1 = 4 \times 3 = 12 \]
This means the smallest dimension of the square that can be formed using the tiles without cutting or overlapping is \(12 , \text{cm}\).
Next, we need to determine the dimensions of the square formed out of these tiles. Since each tile measures \(12 , \text{cm} \times 6 , \text{cm}\), we can arrange the tiles to fill a square of dimension \(12 , \text{cm}\), so each side of the square will be the LCM value that we found, which is \(12 , \text{cm}\).
However, we can also consider larger squares. The smallest square has to be able to accommodate tiles in full rows and columns. The smallest side length of a square must also be a multiple of both \(12\) and \(6\).
The smallest square side length is the least common multiple of \(12\) and \(6\), which we have calculated as \(12\). Since \(12\) itself is the side length, we can have the following possibilities of arranging tiles:
- Using one tile placed in such a way that the \(12 , \text{cm}\) side is horizontal and the \(6 , \text{cm}\) side is vertical, or
- Using \(2\) \(12 cm\) tiles side by side if we scale up to larger possibilities, but they won't be smaller squares.
In conclusion, the smallest square side that can be created from these tiles without any overlapping or cutting is: \[ \boxed{12} \] centimeters.
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