Peter has many cuboid blocks with dimensions 24 cm by 20 cm by 15 cm. He wishes to manufacture a box in the shape of a cube to pack the blocks such that there is no empty space in the box.
Find the smallest possible length of the box. Answer ... cm

1. Recognise that the cube side length must be a common multiple of all three block dimensions.

For the cuboid blocks to fit with no empty space in a cube, the side length of the cube must be divisible by $24$, $20$ and $15$.

So the smallest possible side length is
$\operatorname{LCM}(24,20,15).$

2. Find the LCM of the three dimensions.

Prime factorise:
$24=2^3\times 3,$
$20=2^2\times 5,$
$15=3\times 5.$

Taking the highest powers of each prime,
$\operatorname{LCM}=2^3\times 3\times 5=120.$