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The information provided is that you have smaller cubes that measure 1 cm on either of the three sides. In order words if you lie each cube side by side you would have the length/width/height of the indicate sides.
For a cube made up of 27 smaller cubes, you can determine the length of each side by factoring the entire size provided. That means for a cube made up of 27 smaller cubes you would have;
This means you would have 3 smaller cubes on the length, 3 smaller cubes on the width and 3 smaller cubes on the height.
Therefore, if you now have 150 of the smaller cubes, by factoring this you would have the following possibility;
This means you would have one side (length) 6 cubes, on another side (width) 5 cubes and on the third side (height) 5 cubes.
That means with 150 smaller cubes you could make a cube with the dimensions,
ANSWER:
The largest big cube you could make would be a 6 cm by 5 cm by 5cm big cube