A carpenter cuts the largest possible cube from a rectangular wooden block measuring 60 cm by 45 cm by 30 cm. What volume of wood, in cubic cm, is left over?

A cube inside the block cannot exceed the SMALLEST dimension, so its edge is 30 cm and its volume 27000 cubic cm. The block holds $60\times 45\times$ 30 = 81000 cubic cm, so 81000 - 27000 = 54000 cubic cm is left. The trap 27000 reports the cube that was cut out rather than the offcut; 81000 is the whole block; 77625 takes the edge as 15 cm, the highest common factor of the three dimensions, which is the largest cube that would TILE the block, not the largest that fits in it.