5. Mom bought a box of lump sugar (sugar in cubes). The children first ate the top layer - 77 cubes, then the side layer - 55 cubes, and finally, the front layer. How many sugar cubes are left in the box?
Answer: 300 or 0.

Solution. A box has three dimensions: height, width, and depth. To find out how many cubes are in the top layer, you need to multiply the width by the depth, and for the side layer, multiply the height by the depth. After the top layer is eaten, the height decreases by 1, while the depth remains the same. Therefore, 77 and 55 must be divisible by the original depth of the box. Since the common factors of 77 and 55 are only 1 and 11. If the depth is 1, then nothing is left after the front layer is eaten. If we assume that some sugar is still left, then the depth of the box is 11. Then the width is 77:11=7 cubes, and the height after the top layer is eaten is 55:11=5 cubes. After the side layer is eaten, the width decreases by 1 (becoming 7-1=6 cubes), and after the front layer is eaten, the depth decreases by 1 and becomes 11-1=10. Therefore, the remaining height of the sugar in the box is 5, the width is 6, and the depth is 10 cubes. That is, there are $5 * 6 * 10=300$ cubes of sugar left in the box.